dissertation

Fakultät für Mathematik

DISSERTATION zur Erlangung des akademischen Grades Doctor rerum naturalium (Dr. rer. nat.)

On some Banach Algebra Tools in Operator Theory

vorgelegt von Dipl.-Math. Markus Seidel, geboren am 27. April 1981 in Zwickau, eingereicht am 06. Oktober 2011.

Betreuer: Gutachter:

Prof. Dr. Peter Junghanns Prof. Dr. Bernd Silbermann Prof. Dr. Peter Junghanns Prof. Dr. Vladimir S. Rabinovich

Seidel, Markus On some Banach Algebra Tools in Operator Theory Dissertation, Fakultät für Mathematik Technische Universität Chemnitz, Oktober 2011

To Gloria, Maria and Emilia.

ii

Introduction This work is devoted to sequences A = {An } of operators An with a specific asymptotic structure. Such sequences usually arise from certain approximation methods applied to bounded linear operators. The fundamental tool in the present approach is to take so-called snapshots W t (A), that are operators which capture some aspects of the asymptotics of the sequence A under consideration and which are derived by a transformation of A and a limiting process. We are particularly interested in criteria for the stability of A. Here, A is called stable if there is an N ∈ N such that the operators An are invertible for all n > N and their inverses are uniformly bounded, i.e. supn>N kA−1 n k < ∞. We are further interested in the so-called Fredholm property of sequences A = {An }, a weaker analogon of stability, which provides surprisingly deep connections between the Fredholm properties of the entries An and certain characteristics of their snapshots W t (A), such as their indices, kernels and cokernels. To be a bit more precise, Fredholmness of A means invertibility of its respective coset in a certain quotient algebra, in analogy to the characterization of the operator Fredholm property as invertibility in the Calkin algebra. Moreover, we will derive results on spectral approximation linking the (pseudo)spectra of the An with the (pseudo)spectra of the snapshots W t (A), and describing the convergence of the norms kAn k, kA−1 n k as well as the condition numbers cond(An ). For this, a second essential tool in the present text is given by local principles, which replace the invertibility problem for an element in an algebra by the problem of invertibility of local representatives of this element in a family of (hopefully simpler) local algebras. These Banach algebra techniques and the structure of the outcoming results are not new at all, but possess a long history and have been developed for and applied to many concrete classes of operators and approximation methods. Throughout this text we will try to convey an impression of the diversity of ideas, methods, papers and authors who contributed to their evolution. For the moment we only mention some of the most important milestones: The story has begun with the approach of Silbermann in his groundbreaking paper [83] where the finite section sequences of Toeplitz operators have been studied with the help of two snapshots, and it initiated an evolution which lead to comprehensive monographs such as [12], [56] or [29] among others. In the paper [10] Böttcher, Krupnik and Silbermann collected the local principles of Simonenko, Allan/Douglas and Gohberg/Krupnik, and even presented a framework which permits to relate the norm of an element under consideration to its local norms. The pioneering work on the asymptotics of norms and pseudospectra (at least for the elements in certain sequence algebras which we have in mind) is due to Böttcher [6], [4]. A rather complete and abstract presentation of the whole theory in the case of C ∗ -algebras, the so-called Standard Model, is subject of the monograph [30] of Hagen, Roch and Silbermann. In this setting the Fredholm theory for operator sequences and the splitting phenomenon of their singular values has been studied. After Böttcher [5] introduced the approximation numbers as substitutes for the singular values into that business in order to extend the observation on the splitting phenomenon

iv

INTRODUCTION

for Toeplitz operators on l2 also to the Banach spaces lp , Rogozhin and Silbermann presented a Banach space based Fredholm theory for matrix sequences [75]. A modification of this approach which is applicable to the finite section sequences of convolution type operators on cones, i.e. to a class of sequences of operators acting on infinite dimensional Banach spaces, was presented by Mascarenhas and Silbermann in [50]. Notice that (almost) all of these variants are based on strong or even ∗-strong 1 convergence, and many of them are restricted to sequences A of operators An acting on finite dimensional spaces. Particularly, approximation methods for operators on spaces like l∞ stay out of reach. The infinite dimensional applications require additional assumptions like ind An = 0. The aim of the present text is a generalization and unification which covers most versions of the mentioned algebraic framework for structured operator sequences and extends them to the infinite dimensional case (without additional conditions) and to non-strongly converging (socalled P-strongly converging) sequences. This will put us in a position to close a couple of gaps in the “big picture”, and to include more exotic settings (such as operators on spaces with a sup-norm). We prove several results on a much more abstract level than before, such that they become available not only for certain specific classes of operators but help to tackle many applications at one sweep. Actually, the mentioned P-strong convergence and the machinery behind this notion will prove to be the key which opens the door to the generalization and unification we have in mind. Perhaps, after having read this text the reader joins us in believing that this concept is in impressive harmony with the stability- and Fredholm theory, and is rather the natural choice than an unpretentious substitute for the classical notions of strong convergence and Fredholmness. The first part of the thesis is devoted to basic notions and results for bounded linear operators, to their (N, )-pseudospectra, and at its heart we develop the concept of approximate projections P. This already appeared in [86] and [73] in elementary forms and was thoroughly revised in [63]. Here we we look at these results from a different point of view, contribute some new proofs, and answer a long-standing question on the characterization of the P-Fredholm property, with important consequences. The Fredholm theory for sequences A is presented in Part 2. Condensed to one sentence the outcome is: Much substantial information on the asymptotics of the entries An of an operator sequence A = {An } is stored in the snapshots W t (A) and in the Fredholmness of A. In the 3rd Part we even achieve a characterization of the Fredholm property of sequences belonging to a certain class of sequence algebras solely in terms of their snapshots. This is done via localization. Throughout the development of the general theory we successively apply the achieved results to band-dominated operators and their finite sections, and conversely we employ this prototypic example as a motivation and a guide through the following steps on the abstract level. Notice that the class of band-dominated operators is subject of many publications [42], [62], [61], [60], [57], [70], [63], [44], [59], [58], [69], [15], mainly done by Rabinovich, Roch, Silbermann and Lindner, and has been the engine for the development of the theory on approximate projections P, in a sense. Finally, Part 4 is devoted to several applications. Many of them are well known and have been intensively studied. Nevertheless, having the abstract tools of the previous parts available, we will recover the known results in an impressive conciseness and clarity, and we will be able to harmonize and to extend them into several directions. Parts of the content have already been published or have been submitted for publication in [80],[81], [82] and [78]. 1 Here,

∗ ) converge strongly. (Bn ) is said to converge ∗-strongly, if both (Bn ) and (Bn

Acknowledgements I am deeply grateful to my advisors Peter Junghanns and Bernd Silbermann for their unbounded support, their continuous encouragement and the stable environment they provided me during my studies. Their office doors have always been open for me and I want to thank for the countless conversations that helped to make my ideas converge.

vi

Contents 1 Bounded linear operators on Banach spaces 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Banach spaces, linear operators and Banach algebras . . . . . . . 1.1.2 Fredholm operators . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Auerbach’s lemma and the versatility of projections . . . . . . . 1.1.4 Notions of convergence . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Example: lp -spaces and the finite section method . . . . . . . . . 1.2 Approximate projections and the P-setting . . . . . . . . . . . . . . . . 1.2.1 Substitutes for compactness, Fredholmness and convergence . . . 1.2.2 On approximation methods for linear equations . . . . . . . . . . 1.2.3 Example: lp -spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 The P-dichotomy, or how to grasp Fredholmness in the P-setting 1.3 Geometric characteristics of bounded linear operators . . . . . . . . . . 1.3.1 One-sided approximation numbers and their relatives . . . . . . . 1.3.2 Hilbert spaces and lower singular values . . . . . . . . . . . . . . 1.4 Spectral properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 The (N, )-pseudospectrum . . . . . . . . . . . . . . . . . . . . . 1.4.3 Convergence of sets . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Quasi-diagonal operators . . . . . . . . . . . . . . . . . . . . . . 1.5 Example: operators on lp -spaces . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Band-dominated operators . . . . . . . . . . . . . . . . . . . . . 1.5.2 Rich operators . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 2 2 3 4 6 7 8 8 12 12 14 21 21 24 26 26 26 32 33 34 34 37

2 Fredholm sequences and approximation numbers 2.1 Sequence algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 J T -Fredholm sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Systems of operators . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The approximation numbers of An and the snapshots W t {An } 2.2.3 Regular J T -Fredholm sequences . . . . . . . . . . . . . . . . . 2.2.4 The splitting property and the index formula . . . . . . . . . . 2.2.5 Stability of sequences A ∈ F T . . . . . . . . . . . . . . . . . . . 2.3 A general Fredholm property . . . . . . . . . . . . . . . . . . . . . . . 2.4 Rich sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Example: finite sections of band-dominated operators . . . . . . . . . 2.5.1 Standard finite sections . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Adapted finite sections . . . . . . . . . . . . . . . . . . . . . . .

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39 40 41 43 45 47 48 50 50 53 55 55 58

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viii

CONTENTS

3 Banach algebras and local principles 3.1 Central subalgebras, localization and the KMS property . . . . . . . . 3.2 Localization in sequence algebras . . . . . . . . . . . . . . . . . . . . . 3.2.1 Localizable sequences . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Convergence of norms and condition numbers . . . . . . . . . . 3.2.3 Convergence of pseudospectra . . . . . . . . . . . . . . . . . . . 3.2.4 Rich sequences meet localization . . . . . . . . . . . . . . . . . 3.2.5 Example: finite sections of band-dominated operators . . . . . 3.3 P-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Uniform approximate projections and the P-Fredholm property 3.3.3 P-centralized elements . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 P-strong convergence . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 A look into the crystal ball . . . . . . . . . . . . . . . . . . . .

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61 62 64 64 68 70 72 73 76 76 77 79 80 82

4 Applications 4.1 Quasi-diagonal operators . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Band-dominated operators on Lp (R, Y ) . . . . . . . . . . . . . . . 4.2.1 Standard finite sections . . . . . . . . . . . . . . . . . . . . 4.2.2 Adapted finite sections . . . . . . . . . . . . . . . . . . . . . 4.2.3 Locally compact operators . . . . . . . . . . . . . . . . . . . 4.2.4 Convolution type operators . . . . . . . . . . . . . . . . . . 4.2.5 A remark on the connection to the discrete case . . . . . . 4.3 Multi-dimensional convolution type operators . . . . . . . . . . . . 4.3.1 The operators and their finite sections . . . . . . . . . . . . 4.3.2 The main result for localizable sequences . . . . . . . . . . 4.3.3 Some members of LT . . . . . . . . . . . . . . . . . . . . . 4.3.4 Rich sequences . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Further relatives of band-dominated operators . . . . . . . . . . . . 4.4.1 Band-dominated operators on lp (N, X) . . . . . . . . . . . . 4.4.2 Slowly oscillating operators . . . . . . . . . . . . . . . . . . 4.4.3 Toeplitz operators with continuous symbol . . . . . . . . . 4.4.4 Cross-dominated operators and the flip . . . . . . . . . . . 4.4.5 Quasi-triangular operators . . . . . . . . . . . . . . . . . . . 4.4.6 Triangle-dominated and weakly band-dominated operators . 4.4.7 Toeplitz operators with symbol in Cp + Hp∞ . . . . . . . . . 4.4.8 A larger class of slowly oscillating operators . . . . . . . . .

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83 84 87 87 91 91 92 93 94 94 96 98 102 105 105 106 106 108 109 111 112 113

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Index

115

References

119

Part 1

Bounded linear operators on Banach spaces In the beginning of this first part we recall several elementary definitions and results, particularly the notions of compactness, Fredholmness and convergence in operator algebras. This is, in large parts, just for the sake of completeness, and the informed reader may skip it. Since these classical notions are too rigid for our purposes, we introduce a fabric of generalizations in Section 1.2, based on so-called approximate projections P, and show how to embed the classical setting into this P-framework. The main contributions of the present text to this approach, which was developed in [86], [73] and [63], are given by Theorems 1.14 and 1.27. Section 1.3 is devoted to the discussion of approximation numbers and their relations to other well known geometric characteristics of linear operators like the singular values and furthermore to the injection and surjection modulus. The latter also play an important role in Section 1.4 where we deal with the spectrum of a bounded linear operator and its approximation via pseudospectra.

2

1.1 1.1.1

PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

Preliminaries Banach spaces, linear operators and Banach algebras

In this text we will exclusively deal with complex Banach spaces, that are normed vector spaces over the field C of complex numbers which are complete with respect to their norms. Therefore we drop this word complex and simply call them Banach spaces. As usual, we denote by N, Z and R the sets of natural, integer and real numbers, respectively. Furthermore, Z+ (Z− ) stands for the sets of all non-negative (non-positive) integers. Analogously, we define R+ and R− . For two Banach spaces X, Y the set of all bounded linear operators A : X → Y is denoted by L(X, Y), can be equipped with pointwise defined linear operations (αA + βB)x := αAx + βBx,

A, B ∈ L(X, Y), α, β ∈ C, x ∈ X

and the so-called operator norm kAkL(X,Y) := sup{kAxkY : x ∈ X, kxkX ≤ 1}, and forms a Banach space, as is well known from every text book on functional analysis. In the case X = Y we shortly write L(X) for L(X, Y) and note that the composition of two operators, given by (AB)x := A(Bx) with x ∈ X, turns L(X) into a Banach algebra. Here a Banach space A with another binary operation · : A × A → A, usually called the multiplication in A, is said to be a Banach algebra if (A, +, ·) forms a ring and the multiplication is related to the norm by the following inequality: kABkA ≤ kAkA kBkA

for all A, B ∈ A.

If there is an element I ∈ A which fulfills AI = IA = A for every A ∈ A and which is of norm one then A is referred to as a unital Banach algebra. In this case, an element A ∈ A is invertible from the left (right) if there is a B ∈ A with BA = I (AB = I, respectively), and it is said to be invertible if it is invertible from both sides. The inverse of an invertible element A is uniquely determined and is usually denoted by A−1 . A Banach subalgebra B of a unital Banach algebra A is said to be inverse closed in A if, whenever an element A ∈ B is invertible in A, also its inverse A−1 belongs to B. RecallP that if an element A in a unital Banach algebra has norm less than one then the Neumann series k∈Z+ Ak converges in A and its sum is the inverse of the element I − A. A linear subspace J of A is called an ideal if AJ ∈ J and JA ∈ J for all A ∈ A and all J ∈ J . For a closed ideal J of a Banach algebra A, the set A/J := {A + J : A ∈ A} of cosets A + J := {A + J : J ∈ J }, provided with the operations α(A + J ) + β(B + J ) := (αA + βB) + J ,

(A + J ) · (B + J ) := (AB) + J ,

for all A, B ∈ A and α, β ∈ C, and the norm kA + J kA/J := inf{kA + JkA : J ∈ J }, is a Banach algebra again, referred to as the quotient algebra of A modulo J . Clearly, if A has a unit element I then I + J is a unit in A/J . Analogously, we get the quotient space X/Z of a Banach space X modulo a closed subspace Z.

1.1. PRELIMINARIES

3

Given a Banach space X, the Banach space X∗ := L(X, C) is referred to as the dual space of X and its elements are the so-called linear functionals on X. For A ∈ L(X, Y), the adjoint operator A∗ ∈ L(Y∗ , X∗ ) is defined by the relation (A∗ f )x = f (Ax) for every f ∈ Y∗ and x ∈ X. Notice that kA∗ k = kAk for every A ∈ L(X, Y), and that A is invertible if and only if A∗ is so.

Finally, let p H be a complex vector space with an inner product h·, ·iH such that the norm kxkH := hx, xiX turns H into a complete normed space. Then H is said to be a Hilbert space. This special case of the more general Banach spaces only plays a minor part in this text. Nevertheless, most ideas and results in the theory which is subject of the forthcoming sections firstly appeared in this much more comfortable setting, and many (or even most) applications involve problems in Hilbert spaces. Therefore we will meet them when talking about the roots, former approaches and important applications. We point out that every element y ∈ H defines a bounded linear functional fy ∈ H∗ by the rule fy (x) := hx, yiH for every x ∈ H. The famous Riesz representation theorem even states the converse: Every bounded linear functional f ∈ H∗ is of the form f = fz with some z ∈ H. Moreover, the mapping UH : H → H∗ , y 7→ fy is an isometric and antilinear bijection. In the sense of this identification of Hilbert spaces with their duals one usually introduces the Hilbert adjoint A? of an operator A ∈ L(H1 , H2 ) in a slightly different form, namely as the operator A? ∈ L(H2 , H1 ) which fulfills hAx, yiH2 = hx, A? yiH1

for all x ∈ H1 and all y ∈ H2 .

−1 ∗ In other words we have A? = UH A UH2 with A∗ being the adjoint operator as introduced above 1 for the Banach space setting.

In all what follows we will denote the norms and inner products without the subscripts indicating the respective spaces, if the situation is unambiguous.

1.1.2

Fredholm operators

This section also contains and summarizes material which is well known and can be found (together with proofs) in [25], [12] or in many textbooks on functional analysis. For a bounded linear operator A ∈ L(X, Y) we define its kernel, its range and its cokernel by ker A := {x ∈ X : Ax = 0}

im A := A(X) = {Ax : x ∈ X}

coker A := Y/ im A.

The kernel of a bounded linear operator A always forms a closed subspace of X, and its range is a subspace of Y but, in general, it is not necessarily closed. We say that A is normally solvable if im A is a closed subspace of Y. One usually refers to rank A := dim im A as the rank of A. Further, A ∈ L(X, Y) is called compact if the closure of the set {Ax : x ∈ X, kxk ≤ 1} in Y is compact. Denote the set of all compact operators A ∈ L(X, Y) by K(X, Y). In case Y = X we again abbreviate K(X, X) by K(X). It is well known that K(X) forms a closed ideal in L(X), and the quotient algebra L(X)/K(X) is usually referred to as the Calkin algebra. The adjoint operator A∗ is compact if and only if A is compact. Similarly, A∗ is normally solvable if and only if A is so, and in this case dim ker A∗ = dim coker A,

dim coker A∗ = dim ker A.

Definition 1.1. An operator A ∈ L(X, Y) is called Fredholm operator if dim ker A < ∞ and dim coker A < ∞. If A is Fredholm then the integer ind A := dim ker A − dim coker A is referred to as the index of A.

4


In the following theorems we record the most important properties of the class of Fredholm operators and several equivalent characterizations of the Fredholm property. Actually, the subsequent sections will reveal that these characterizations constitute a corner stone and a guide throughout the whole text. Theorem 1.2. Let Φ(X, Y) denote the set of all Fredholm operators in L(X, Y). Then: • Φ(X, Y) is an open subset of L(X, Y) and for A ∈ Φ(X, Y) we have dim ker(A + C) ≤ dim ker A, dim coker(A + C) ≤ dim coker A whenever C has sufficiently small norm. The mapping ind : Φ(X, Y) → Z is constant on the connected components of Φ(X, Y). • Let A ∈ Φ(X, Y) and K ∈ K(X, Y). Then A + K ∈ Φ(X, Y) and ind(A + K) = ind A. • (Atkinson’s theorem). Let Z be a further Banach space, A ∈ Φ(Y, Z) and B ∈ Φ(X, Y). Then we have AB ∈ Φ(X, Z) and ind(AB) = ind A + ind B. Theorem 1.3. Let A ∈ L(X, Y). Then the following conditions are equivalent: 1. A is Fredholm. 2. A is normally solvable, dim ker A < ∞, and dim ker A∗ < ∞. 3. A∗ is Fredholm. 4. There exist projections

1

P ∈ K(X) and P 0 ∈ K(Y) s.t. im P = ker A and ker P 0 = im A.

5. The following is NOT true: For each l ∈ N and each > 0 there exists a projection Q ∈ K(X) or a projection Q0 ∈ K(Y) with rank Q, rank Q0 ≥ l such that kAQk < or kQ0 Ak < . In case Y = X the list can be completed by 6. The coset A + K(X) is invertible in the Calkin algebra L(X)/K(X). Proof. The equivalence of the Assertions 1. till 3. and 6. is well known, and can again be found in the mentioned textbooks. For 4. ⇒ 1. notice that compact projections are of finite rank. The role of compact projections is subject of the next section and the considerations there provide the proof of the remaining implications. In particular, 2. ⇒ 4., 5. as well as NOT 2. ⇒ NOT 5., and hence 5. ⇒ 2., follow from Proposition 1.6.

1.1.3

Auerbach’s lemma and the versatility of projections

One of the most valuable instruments in this work is the characterization of subspaces via projections, that are operators P ∈ L(X) with P 2 = P . To be more precise, we say that a closed subspace X1 of a Banach space X is complemented if there is another closed subspace X2 ⊂ X (a so-called complement of X1 ) such that X decomposes into X1 and X2 in the sense that X = X1 + X2 and X1 ∩ X2 = {0}. In this case, every x ∈ X admits a unique decomposition x = x1 + x2 with xi ∈ Xi , i = 1, 2. It is well known that the mapping P : x 7→ x1 is a (bounded) projection onto X1 and I − P is a (bounded) projection onto X2 . One further says that P maps parallel to X2 . Conversely, for every projection P ∈ L(X) the space ker P is complemented and im P is a complement. 1 An

operator P ∈ L(X) is called projection if P 2 = P .

1.1. PRELIMINARIES

5

Proposition 1.4. (Auerbach’s lemma 2 ) Let X be of dimension n. Then there exist {xi }ni=1 ⊂ X and functionals {fi }ni=1 ⊂ X∗ such that, for every i, k, ( 1 :i=k . kxi k = kfk k = 1 and fk (xi ) = 0 : i 6= k This immediately yields the existence of bounded projections onto finite dimensional and finite codimensional subspaces of Banach spaces. For details see [54], B.4.9f. Proposition 1.5. Let X be a Banach space. If X1 ⊂ X is an m-dimensional subspace then there is a projection P onto X1 of norm kP k ≤ m. If X2 is a subspace having codimension n then there is a projection P 0 parallel to X2 of norm kP 0 k ≤ n + 1. Now we can capture all Fredholm properties of bounded linear operators in terms of projections, as the next well known proposition shows. For completeness we give its short proof. Proposition 1.6. Let X, Y be Banach spaces and A ∈ L(X, Y). 1. If A is normally solvable and k ≤ dim ker A (k ≤ dim coker A) then there is a projection P ∈ L(X) (P 0 ∈ L(Y)) of rank k and with norm less than k+2 such that AP = 0 (P 0 A = 0, respectively). 2. If A is not normally solvable then, for every k ∈ N and every > 0, there are projections P ∈ L(X) and P 0 ∈ L(Y) of rank k and with norm less than k + 2 such that kAP k < and kP 0 Ak < . Proof. In order to prove the second assertion, assume first that im A is not closed. We want to show that for every l ∈ N and for every δ > 0 there exists an l-dimensional subspace Xl of X such that kA|Xl k < δ. Fix δ > 0. Then there exists x1 ∈ X, kx1 k = 1 such that kAx1 k < δ, otherwise the range of A would be closed. Assume that the assertion is true for l − 1. By Proposition 1.5 there exist a complement Z of Xl−1 and a projection Q onto Xl−1 with ker Q = Z and kQk ≤ l. A(Z) is not closed (otherwise A(X) = A(Xl−1 ) + A(Z) would be the sum of a finite dimensional and a closed subspace of Y and hence closed, as is known). Now we can choose xl ∈ Z with kxl k = 1 and kAxl k < δ, and for arbitrary scalars α, β and elements x ∈ Xl−1 we see kA(αx + βxl )k ≤ kA(αx)k + kA(βxl )k < δ(kαxk + kβxl k)

kαxk = kQ(αx)k = kQ(αx + βxl )k ≤ kQkkαx + βxl k

kβxl k = k(I − Q)(βxl )k = k(I − Q)(αx + βxl )k ≤ (1 + kQk)kαx + βxl k,

i.e. kA(αx + βxl )k < δ(1 + 2l)kαx + βxl k. Since δ > 0 is arbitrary the assertion follows by induction. Consequently, for k ∈ N and > 0 we can choose 0 < δ < /k, a k-dimensional subspace of X and an appropriate projection P onto this subspace with kP k ≤ k, such that kAP k ≤ δkP k < . To find P 0 in the second assertion we recall that A∗ is not normally solvable whenever A is so, by what we have hence there is a projection Q ∈ L(Y∗ ) of rank k such that kA∗ Qk < k+2 ˆ be the intersection of the kernels of all functionals in im Q. Obviously, proved before. Let Y ˆ has codimension k and we can choose a rank k projection P 0 of norm less than k + 2 (see Y ˆ Then f ◦ (I − P 0 ) = 0 for all f ∈ im Q, that is Proposition 1.5) such that I − P 0 is onto Y. 2 For

a proof see [54], B.4.8.

6


(I − P 0 )∗ Q = 0. Since (P 0 )∗ and Q are projections of the same (finite) rank we conclude that im(P 0 )∗ = im Q and kP 0 Ak = kA∗ (P 0 )∗ k = kA∗ Q(P 0 )∗ k ≤ kA∗ QkkP 0 k < .

Now, for the first assertion apply Proposition 1.5 to find P . To construct P 0 , recall that dim ker A∗ and dim coker A coincide in case of a normally solvable operator A, choose a rank-k-projection Q onto a subspace of ker A∗ , and proceed as above. Also the notion of generalized invertibility can be completely characterized in terms of projections. Remember that an operator A ∈ L(X, Y) is said to be generalized invertible, if there is an operator B ∈ L(Y, X) such that A = ABA and B = BAB. Proposition 1.7. Let A ∈ L(X, Y) and B ∈ L(Y, X). Then the following conditions are equivalent • ABA = A. • IX − BA is a bounded projection with im(IX − BA) = ker A. • IY − AB is a bounded projection with ker(IY − AB) = im A. Moreover, if A ∈ L(X, Y) is an operator and P ∈ L(X), P 0 ∈ L(Y) are projections with im P = ker A and ker P 0 = im A then there exists an operator C ∈ L(Y, X) with A = ACA, C = CAC and P = IX − CA, P 0 = IY − AC. Proof. Let ABA = A. Then (IX − BA)2 = IX − BA − BA + BABA = IX − BA − BA + BA = IX − BA and

(IY − AB)2 = IY − AB − AB + ABAB = IY − AB − AB + AB = IY − AB are projections which fulfill

im A = im ABA ⊂ im AB ⊂ im A, hence im A = im AB = ker(IY − AB), and

ker A ⊂ ker BA ⊂ ker ABA = ker A, hence ker A = ker BA = im(IX − BA). The rest of the first part is obvious.

Now let A, P and P 0 be given as stated. Then ker P and ker P 0 are closed. As a consequence of the Banach inverse mapping theorem, the operator A|ker P : ker P → ker P 0 is invertible. Let A(−1) be its inverse. Then C := (I − P )A(−1) (I − P 0 ) does the job.

1.1.4

Notions of convergence

Definition 1.8. A sequence (An ) of operators An ∈ L(X, Y) is said to converge to A ∈ L(X, Y) • strongly if kAn x − AxkY → 0 for each x ∈ X (we write A = s-lim An ), n→∞

• uniformly if kAn − AkL(X,Y) → 0. Let (An ) ⊂ L(X, Y) be a bounded sequence of operators. The Banach Steinhaus Theorem states that if An x converges for every x ∈ X then Ax := limn An x defines a bounded linear operator A ∈ L(X, Y), and kAk ≤ lim inf kAn k. n→∞

It is well known that compact operators turn strong convergence into uniform convergence:

1.1. PRELIMINARIES

7

Proposition 1.9. Let X be a Banach space and A, An ∈ L(X). Then • An → A strongly ⇔ An K → AK uniformly for every K ∈ K(X). • A∗n → A∗ strongly ⇒ KAn → KA uniformly for every K ∈ K(X). We say that a sequence of operators An converges ∗-strongly, if (An ) as well as (A∗n ) converge strongly.

1.1.5

Example: lp -spaces and the finite section method

Now, it’s time to illustrate the introduced notions and their interactions with an example. Let X stand for a fixed complex Banach space and, for 1 ≤ p < ∞, let lp = lp (Z, X) denote the Banach space of all sequences x = (xi )i∈Z of elements xi ∈ X such that 3 X kxkplp := kxi kpX < ∞. (1.1) i∈Z

Note that the linear structure is given by entrywise defined addition and scalar multiplication, and (1.1) defines the norm. We further introduce l∞ = l∞ (Z, X) as the Banach space of all x = (xi ) with kxkl∞ := sup kxi kX < ∞. i∈Z

In all what follows we again omit the subscript of the norm if the situation is unambiguous. For 1 ≤ p < ∞ the dual space of lp (Z, X) can be identified with lq (Z, X ∗ ) where 1/p + 1/q = 1. Here, the dual product is given by X h(xi ), (yi )i := yi (xi ). i∈Z

Let, for a moment, p = 2 and X be a Hilbert space of finite dimension. ThenPl2 becomes a separable Hilbert space, provided with the scalar product defined by h(xi ), (yi )i := i∈Z hxi , yi iX . Moreover, we can introduce projections Pm ∈ L(l2 ), m ∈ N, by the rule Pm : (xi ) 7→ (. . . , 0, x−m , . . . , xm , 0, . . .)

(1.2)

and we may easily check that the sequence (Pm )m∈N converges ∗-strongly to the identity. For an equation Ax = b with given A ∈ L(l2 ), b ∈ l2 and the unknown x ∈ l2 , the so-called finite section method uses the (finite dimensional) linear equations Pn APn xn = Pn b as substitutes for Ax = b. One hopes that they provide solutions xn (at least for sufficiently large n) which converge to the desired x. The following result, which is due to Polski [55], justifies this approach. Proposition 1.10. Let X be a Banach space and suppose that A ∈ L(X) is invertible, the sequence (An ) ⊂ L(X) converges strongly to A, (bn ) ⊂ X converges to b ∈ X, and (An ) is stable, that is ∃ N such that An is invertible for n > N and sup kA−1 n k < ∞. n>N

Then An xn = bn are uniquely solvable for n > N and the solutions xn converge to the (unique) solution x of Ax = b. 3 One

usually says that the sequence x is p-summable.

8


So, we see that for an invertible A the stability of the finite section sequence (Pn APn ) provides the applicability of the finite section method to approximate the solution of Ax = b.4 It is well known that the projections Pm generate the ideal K(l2 ) of all compact operators in the sense, that the smallest closed ideal in L(l2 ) which contains all Pm coincides with K(l2 ). This observation conveys an impression that also compact operators will occur in the treatment of the stability, and, indeed, it has turned out during the last decades that one has to deal with ideals of “compact” sequences of the form 5 {(Pn KPn ) + (Gn ) : K ∈ K(l2 ), kGn k →n→∞ 0}. Now, we want to relax our assumptions on p and X and we ask what happens if 1 ≤ p ≤ ∞ and X is an arbitrary Banach space. The definition of the Pn extends correctly to all cases, the Polski result covers all spaces lp with 1 ≤ p < ∞, but p = ∞ stays out of reach, because there the sequence (Pn ) does not converge strongly anymore. Since the classical approaches for the stability even require ∗-strong convergence in general, also the case p = 1 is unattainable. One reason for this is the fact that in the cases p ∈ {1, ∞} the ideal which is generated by the operators Pn does not coincide with the ideal of the compact operators anymore. Moreover, since the projections Pn are non-compact if dim X = ∞, the picture also gets dramatically worse in the infinite dimensional case. Therefore, the next section is devoted to a slightly modified framework, which will again provide certain notions of compactness, Fredholmness and convergence having similar properties and interactions, which will allow to restate and to extend the Polski result and the theory which is necessary to characterize the stability, but which is more flexible. In particular, we aspire to an equal treatment of all cases 1 ≤ p ≤ ∞ and all spaces X.

1.2

Approximate projections and the P-setting

Looking back at the development and also at the modern proofs in the Banach algebra techniques which are used for the theory of approximation methods it becomes apparent that the interactions between compactness, Fredholmness, and strong convergence 6 constitute a large part of the core. We now want to modify this triple such that these fruitful relations survive and in this way also the familiar proof strategies still work, but a larger flexibility is achieved, such that we can also cover approximation methods which do not converge strongly. In a way, we turn the table and we no longer hunt for methods which suit to the classical notions of compactness, Fredholmness and convergence, but we take the method itself as the starting point and construct a suitable triple.

1.2.1

Substitutes for compactness, Fredholmness and convergence

Approximate projections Let X be a Banach space and let P = (Pn )n∈N be a bounded sequence of operators in L(X) with the following properties: • Pn 6= 0 and Pn 6= I for all n ∈ N, • For every m ∈ N there is an Nm ∈ N such that Pn Pm = Pm Pn = Pm if n ≥ Nm . 4 Notice that for an operator A ∈ L(lp ) the finite section sequence (P AP ) is stable (where the operators n n Pn APn are considered as operators in L(im Pn ), respectively) if and only if the sequence (An ) ⊂ L(lp ) with An = Pn APn + (I − Pn ) is stable. 5 See, for example, the pioneering paper [83] of Silbermann, the monograph [12] or our applications in Part 4. 6 See Theorem 1.3 and Proposition 1.9

1.2. APPROXIMATE PROJECTIONS AND THE P-SETTING

9

Then P is referred to as an approximate projection.7 In all what follows we set Qn := I − Pn and we write m n if Pk Ql = Ql Pk = 0 for all k ≤ m and all l ≥ n. P-compactness Let P be an approximate projection. A bounded linear operator K is called P-compact if kKPn − Kk and kPn K − Kk tend to zero as n → ∞. By K(X, P) we denote the set of all P-compact operators on X and by L(X, P) the set of all bounded linear operators A for which AK and KA are P-compact whenever K is P-compact. Theorem 1.11. Let P be an approximate projection on the Banach space X. L(X, P) is a closed subalgebra of L(X), it contains the identity operator, and K(X, P) is a proper closed ideal of L(X, P). An operator A ∈ L(X) belongs to L(X, P) if and only if, for every k ∈ N, kPk AQn k → 0 and kQn APk k → 0 as n → ∞. The “P-concept” which we are introducing here will be developed later on in Part 3 also in more abstract Banach algebras. Therefore we will elaborate the details there and, for the moment, we refer to Theorem 3.25. P-Fredholmness and invertibility at infinity Here are two possible generalizations of Fredholmness based on P-compact operators instead of compact ones which are motivated by Theorem 1.3. Definition 1.12. We say that A ∈ L(X) is invertible at infinity (with respect to P) if there is an operator B ∈ L(X) with I − AB, I − BA ∈ K(X, P). In this case B is referred to as a P-regularizer for A. An operator A ∈ L(X, P) is said to be P-Fredholm if the coset A + K(X, P) is invertible in the quotient algebra L(X, P)/K(X, P). Notice that invertibility at infinity is defined in L(X), whereas for P-Fredholmness we are restricted to L(X, P), since we need that K(X, P) forms a closed ideal in L(X, P). Both notions have been known and have been studied for a long time. The monographs by Rabinovich, Roch and Silbermann [63] and Lindner [44] already contain a comprehensive theory and many applications of this approach. But until now it was an open problem if these two notions coincide for operators A ∈ L(X, P). Here we answer this question affirmatively, under a natural condition on P which has been in the business since the inverse closedness of L(X, P) in L(X) is known, based on a proof of Simonenko [86]. Definition 1.13. Given a Banach space X with an approximate projection P = (Pn ), we set S1 := P P1 and Sn := Pn − Pn−1 for n > 1. Further, for every finite subset U ⊂ N, we define PU := k∈U Sk . P is said to be uniform if CP := sup kPU k < ∞, the supremum over all finite U ⊂ N. Theorem 1.14. Let P be a uniform approximate projection on X. An operator A ∈ L(X, P) is P-Fredholm if and only if it is invertible at infinity. In this case every P-regularizer of A belongs to L(X, P). Particularly, L(X, P) is inverse closed in L(X). This main result (except the inverse closedness) is really new and due to the author, closes a long-standing gap in the theory of approximate projections and has been published in [81] for the first time. It is a special case of Theorem 3.28 whose proof is essentially based on Theorem 3.27 which is new as well, interesting on its own and, in the present context, reads as follows. 7 This

notion was introduced in [73].

10


Theorem 1.15. Let P be a uniform approximate projection on X and A ∈ L(X, P). Then there is an equivalent uniform approximate projection Pˆ = (Fn ) on X (depending on A) with CPˆ ≤ CP such that k[Fn , A]k = kAFn − Fn Ak → 0 as n → ∞.

Here, two approximate projections P = (Pn ) and Pˆ = (Fn ) on X are said to be equivalent if for every m ∈ N there is an n ∈ N such that Pm Fn = Fn Pm = Pm

and Fm Pn = Pn Fm = Fm .

ˆ Indeed, for K ∈ K(X, P), Note that in this case we have K(X, P) = K(X, P). kK(I − Fn )k ≤ kKQm kkI − Fn k + kKkkPm (I − Fn )k, where the first summand tends to zero as m → ∞ and, for every fixed m, the second one equals zero for an n0 by the definition, and then also for all n Pˆ n0 . Analogously, we check ˆ k(I − Fn )Kk → 0 as n → ∞, hence K ∈ K(X, P). Reversing the roles of P and Pˆ gives ˆ the claim. Consequently, also L(X, P) = L(X, P) and the notions of P-compactness and Pˆ Fredholmness coincide with the respective P-notions. The same holds true for the so-called P-strong convergence, a substitute for the third part of the triple, the strong convergence, which we introduce in the next paragraph. P-strong convergence Let P = (Pn ) be an approximate projection and, for each n ∈ N, let An ∈ L(X). The sequence (An ) converges P-strongly to A ∈ L(X) if, for all K ∈ K(X, P), both k(An − A)Kk and kK(An − A)k tend to 0 as n → ∞. In this case we write An → A P-strongly or A = P-limn An . We mention that this definition is based on the similar properties of strong convergence as stated in Proposition 1.9 and was introduced in [73]. Proposition 1.16. A bounded sequence (An ) in L(X) converges P-strongly to A ∈ L(X) iff k(An − A)Pm k → 0 and kPm (An − A)k → 0 for every fixed Pm ∈ P.

(1.3)

This easily follows from P ⊂ K(X, P) and the estimates k(An − A)Kk ≤ kKkk(An − A)Pm k + kAn − AkkQm Kk

kK(An − A)k ≤ kKkkPm (An − A)k + kAn − AkkKQm k. Unfortunately, the P-strong limit is not unique in general. Consider, for instance, a projection P ∈ / {0, I} and P = (Pn ) given by Pn := P for all n. Then the sequence (Pn ) converges P-strongly to both P and I. Therefore we adopt a further condition on P from [63] to force uniqueness. Definition 1.17. An approximate projection P is called approximate identity if for every x ∈ X the estimate supn kPn xk ≥ kxk holds. By F(X, P) we denote the set of all bounded sequences (An ) ⊂ L(X), which possess a P-strong limit in L(X, P). Remark 1.18. Let P be an approximate identity. Then kAk ≤ supm kPm Ak holds for every bounded linear operator A. To check this, fix > 0 and choose x0 ∈ X, kx0 k = 1 such that kAx0 k ≥ kAk − . Furthermore, take n0 such that kPn0 Ax0 k ≥ kAx0 k − to get kAk ≤ kPn0 Ax0 k + 2 ≤ kPn0 Ak + 2 ≤ sup kPm Ak + 2. m∈N

This gives the claim, since was chosen arbitrarily.


11

Besides the uniqueness of P-strong limits this provides several further structural properties. Theorem 1.19. Let P be an approximate identity on X. Then

8

• The algebra L(X, P) is closed with respect to P-strong convergence, this means if a sequence (An ) ⊂ L(X, P) converges P-strongly to A then A ∈ L(X, P). Moreover, (An ) is bounded in this case, and hence belongs to F(X, P). • The P-strong limit of every (An ) ∈ F(X, P) is uniquely determined. • Provided with the operations α(An ) + β(Bn ) := (αAn + βBn ), (An )(Bn ) := (An Bn ), and the norm k(An )k := supn kAn k, F(X, P) becomes a Banach algebra with identity I := (I). The mapping F(X, P) → L(X, P) which sends (An ) to its limit A = P-limn An is a unital algebra homomorphism and (with BP := supn kPn k) kAk ≤ BP lim inf kAn k. n→∞

(1.4)

Proof. For the details of the proof we again refer to the more general result given with Theorem 3.34 and note that we can set DP := BP due to Remark 1.18 and Remark 3.33. Some remarks on inclusions It is obvious that for arbitrary Banach spaces X with an approximate projection P the inclusions K(X, P) ( L(X, P) ⊂ L(X) hold. To disclose how the set of compact operators fits into that picture we borrow the following result from [44], Proposition 1.24. Proposition 1.20. The inclusion K(X, P) ⊂ K(X) holds if and only if P ⊂ K(X). If P is an approximate identity on X then we also have L(X, P) ∩ K(X) ⊂ K(X, P). Proof. At first, recall that every operator A ∈ K(X, P) is the norm limit of the sequence (APn ). Thus it is compact if all APn are compact. If, conversely, one Pm is not compact, then it belongs to K(X, P) \ K(X). Now let P be an approximate identity, let A ∈ L(X, P) be compact, and assume that kAQn k does not tend to zero. Then there is a bounded sequence (xn ) ⊂ X such that AQn xn does not tend to zero as well, but with the help of Theorem 1.11 we see that at least Pk AQn xn → 0 as n → ∞ for every fixed k. Since A is compact, it is always true that each subsequence of (AQn xn ) has a convergent subsequence and the respective limit y fulfills Pk y = 0 for every k, hence y = 0 since P is an approximate identity. This contradicts AQn xn 6→ 0. To show that also kQn Ak → 0, we mention that kQn Ak ≤ kQn APk k + kQn kkAQk k, where the last term at the right hand side can be made as small as desired by choosing k large and the first one tends to zero for every fixed k by Theorem 1.11. 8 These

results are already known from [63], Section 1.1.4 et seq.

12


1.2.2

On approximation methods for linear equations

Let X be a Banach space with an approximate projection P. In accordance with the definition for operator sequences we say that a sequence (xn ) of elements xn ∈ X converges P-strongly to x ∈ X if kK(xn − x)k → 0 as n → ∞, for every K ∈ K(X, P). It is obvious from the definition that a bounded sequence (xn ) ⊂ X converges P-strongly, iff kPm (xn − x)k → 0 as n → ∞,

for every

m ∈ N.

Definition 1.21. A Banach space X with an approximate projection P is called complete w.r.t. P-strong convergence if the following holds: Every bounded sequence (xn ) ⊂ X for which (Pm xn )n∈N is a Cauchy sequence for every m ∈ N has a P-strong limit x ∈ X. Here comes the announced generalized version of the Polski result (Proposition 1.10) which was discovered by Roch and Silbermann in [73]. Proposition 1.22. Let P be a uniform approximate projection on the Banach space X. Suppose that (bn ) ⊂ X is bounded and converges P-strongly to b ∈ X, A ∈ L(X, P) is invertible, and the sequence (An ) ⊂ L(X) is stable and converges P-strongly to A. Then the solutions xn of An xn = bn converge P-strongly to the solution x of Ax = b. Proof. Let K ∈ K(X, P), n be sufficiently large such that An is invertible, and consider −1 −1 kK(xn − x)k = kK(A−1 b)k = kK((A−1 )bn + A−1 (bn − b))k n bn − A n −A −1 = kKA−1 (A − An )A−1 (bn − b)k n bn + KA

−1 ≤ kKA−1 (A − An )kkA−1 (bn − b)k. n bn k + kKA

The operator A−1 belongs to L(X, P) by Theorem 1.14, hence KA−1 is P-compact. This obviously yields that kKA−1 (A − An )k and kKA−1 (bn − b)k tend to zero as n → ∞. Since kA−1 n bn k are uniformly bounded w.r.t. n, the assertion follows.

1.2.3

Example: lp -spaces

Let us again rest for a minute and ask, what this means for the spaces lp = lp (Z, X) which were introduced above in Section 1.1.5 and for the finite section method applied to bounded linear operators on these spaces. First of all we note that the sequence P = (Pn ) as defined in (1.2) gives a uniform approximate identity with BP = CP = 1 for all 1 ≤ p ≤ ∞ and all Banach spaces X. Therefore all results of Section 1.2 can be applied. That is, the algebra L(lp , P) is closed under passing to inverses or P-regularizers (by Theorem 1.14), the notions “P-Fredholm” and “invertible at infinity” are equivalent, and P-strong convergence is well defined (see Theorem 1.19). We will even see that Fredholm operators in L(lp , P) are automatically P-Fredholm (Corollary 1.28). Figure 1.1 illustrates the relations between the respective operator algebras L(lp ), K(lp ), L(lp , P), and K(lp , P) for the various configurations of p and dim X. To substantiate the asserted proper inclusions there, we consider two examples which we again take from [44]. • Let a ∈ X and f ∈ X ∗ be non-zero elements and let A ∈ L(l1 ) be given by the rule ! X f (xi )a, 0, 0, . . . . A : (xi ) 7→ . . . , 0, 0, i


13

dim X = ∞

dim X < ∞ L(lp ) = L(lp , P)

L(lp ) .I

.B

L(lp , P)

.I

K(lp , P) .Pn

1 0 is constant. Thus, A is not P-deficient, hence properly n k) P-Fredholm. Suppose that the kernel of A is not trivial. Then there is a non-trivial projection P ∈ K(X, P), P 6= 0 such that 0 = kAP k ≥ CkP k ≥ C, yielding a contradiction. Thus A is injective. Analogously one shows that A is surjective and hence invertible, due to the Banach inverse mapping theorem. From Theorem 1.14 we even get A−1 ∈ L(X, P). Further, we know for large n and every K ∈ K(X, P) that −1 −1 −1 (A−1 )K = A−1 )K = A−1 K, n −A n (I − An A n (A − An )A

which obviously converges to zero in the norm as n → ∞. In the same way we find that −1 −1 follows. ) → 0 in the norm, hence the P-strong convergence A−1 K(A−1 n →A n −A Proofs and technical details Let us try to gather more understanding of the somewhat mysterious P-dichotomy. First of all, from Proposition 1.7 we immediately conclude Corollary 1.30. An operator A ∈ L(X) is properly P-Fredholm iff there is a P-regularizer B ∈ L(X) for A which is also a generalized inverse, that is I − AB, I − BA ∈ K(X, P) and ABA = A, BAB = B. If A ∈ L(X, P) and P is even uniform then the situation becomes more relaxed. Corollary 1.31. Let P be a uniform approximate projection. Then A ∈ L(X, P) is properly P-Fredholm if and only if it is P-Fredholm and has a generalized inverse in L(X, P). In this case there is a B ∈ L(X, P) which is P-regularizer and generalized inverse at the same time. Proof. Let A ∈ L(X, P) be properly P-Fredholm. Then, by Corollary 1.30, there is an operator B ∈ L(X) such that A = ABA, B = BAB and I − AB, I − BA ∈ K(X, P). From Theorem 1.14 we obtain that B ∈ L(X, P). Conversely, let A be P-Fredholm with P-regularizer C ∈ L(X, P) and let B ∈ L(X, P) be a generalized inverse for A. The projections P := I − BA, P 0 := I − AB are contained in L(X, P). Moreover, P = (I − CA)P + CAP = (I − CA)P and P 0 = P 0 (I − AC) + P 0 AC = P 0 (I − AC) even show that P, P 0 ∈ K(X, P), that is B is a P-regularizer for A. In view of Corollary 1.30, this yields the proper P-Fredholmness.

16


Proposition 1.26 immediately results from the following observation. Proposition 1.32. Let A ∈ L(X) be invertible at infinity (w.r.t. an approximate projection P). If A is normally solvable then for every k ∈ N with k ≤ dim ker A (k ≤ dim coker A) there is a P-compact projection P of rank k such that AP = 0 (P A = 0). If A is not normally solvable then A is properly P-deficient from both sides. Proof. Let B ∈ L(X) be a P-regularizer for the operator A. For every x ∈ ker A we find that Qn x = Qn (I − BA)x + Qn BAx = Qn (I − BA)x tends to zero as n → ∞. Let X1 be a finite dimensional subspace of ker A. Then we can fix m ∈ N s.t. sup{kQm xk : x ∈ X1 , kxk = 1} < 1/2 and deduce that the operator Pm : X1 → X2 := Pm (X1 ) is invertible and its inverse has norm less than 2. Let S denote its inverse and let m ˜ m. Since X2 is a finite dimensional subspace of the Banach space ker Qm ˜ , there is a bounded projection R ∈ L(ker Qm ˜ ) onto X2 with kRk ≤ dim X1 , by Proposition 1.5. Now, we define P := SRPm and we easily check that im P = X1 as well as P 2 = SRPm P = SPm P = P , hence P is a projection onto X1 which obviously belongs to K(X, P). Assume now that A is not normally solvable and fix > 0 and k ∈ N. Proposition 1.6 provides us with a rank-k-projection T such that kAT k ≤ . Further, denote by d the finite number supn kQn k and check that kQn T k ≤ kQn (I − BA)T k + kQn BAT k

≤ kQn (I − BA)kkT k + dkBk ≤ 2dkBk

for sufficiently large n. Further, for x ∈ im T , kAxk + kAkkQn xk 1 + 2dkBkkAk kAPn xk ≤ ≤ . kPn xk kxk − kQn xk 1 − 2dkBk This shows that for sufficiently small and sufficiently large l the space X3 := im Pl T is of dimension k and kAzk ≤ 4dkBkkAkkzk holds for all z ∈ X3 . Since X3 ⊂ ker Q˜l ⊂ ker Qˆl for ˆl ˜l l we can again choose a projection R ∈ L(ker Qˆ) onto X3 with norm at most k and l define P := RP˜l . Obviously, P is a P-compact projection of rank k and we have the estimate kAP k ≤ 4kd(d + 1)kBkkAk. Since was chosen arbitrarily, this proves the proper P-deficiency from the right. Further, if A is not normally solvable, then A∗ is not normally solvable, too, and for every prescribed > 0 we find a projection T ∈ K(X∗ , P ∗ ) of rank k with kA∗ T k ≤ . We show that there is a projection P 0 ∈ K(X, P) of norm less than 2k(k + 1)(d + 1) such that im(P 0 )∗ = im T , and hence kP 0 Ak = kA∗ (P 0 )∗ k = kA∗ T (P 0 )∗ k ≤ 2k(k + 1)(d + 1). Then, since was chosen arbitrarily, A proves to be properly P-deficient from the left. For this, let l be such that kQ∗l T k < 1/2. Then, for every f ∈ im T we have kf ◦ Pl k = kPl∗ f k ≥ kf k − kQ∗l f k = kf k − k(Q∗l T )f k ≥

1 kf k, 2

(1.5)

that is {f |im Pl : f ∈ im T } forms a linear space of dimension k. Hence, for ˜l l, \ X1 := ker f |ker Ql˜ ⊂ ker Q˜l f ∈im T

has codimension k in ker Q˜l . Due to Proposition 1.5 we can choose a projection R parallel to X1 onto a certain complement X2 of X1 in ker Q˜l with the norm not greater than k + 1. Since the


17

set of all restrictions g = f |X2 of the functionals f ∈ im T to X2 forms a k-dimensional space we conclude that each functional on X2 is of the form g = f |X2 with f ∈ im T . Auerbach’s Lemma (Proposition 1.4) provides bases x1 , . . . , xk ∈ X2 and f1 , . . . , fk ∈ im T such that kxi k = kgj k = 1 and gj (xi ) = δij for all i, j = 1, . . . , k, where gj := fj |X2 (here δij = 1 if i = j, and δij = 0 otherwise). Due to (1.5) the norms kfj k can be estimated by kfj k ≤ 2kfj ◦ Pl k ≤ 2kfj ◦ R ◦ Pl k + 2kfj ◦ (I − R) ◦ Pl k ≤ 2kgj kkRkkPl k < 2(k + 1)(d + 1), Pk thus, defining P 0 x := i=1 fi (x)xi , we obtain a P-compact projection onto X2 with the norm Pk less than 2k(k + 1)(d + 1). Since fj (P 0 x) = i=1 fi (x)fj (xi ) = fj (x) for all j and x we find that im(P 0 )∗ = im T . It remains to consider operators A which are normally solvable, hence dim coker A = dim ker A∗ , and to check that for every finite k ≤ dim coker A there is a P-compact projection P 0 of rank k with P 0 A = 0. Since A∗ is invertible at infinity (w.r.t. P ∗ ) we can apply the first part of this proof to find a P ∗ -compact projection T of rank k onto a respective subspace of ker A∗ . Then we simply apply the above argument with = 0 to construct P 0 . For the proof of Theorem 1.27 we need some preliminary results and we introduce a useful and important closed subspace X0 of X by X0 := {x ∈ X : kQn xk → 0 as n → ∞}. Proposition 1.33. . 1. For each A ∈ L(X, P) the restriction A|X0 to X0 is contained in L(X0 , P) and if K ∈ K(X, P) then K|X0 ∈ K(X0 , P). 2. Let X1 ⊂ X0 be a finite dimensional subspace. Then there is a projection P ∈ K(X, P) onto X1 with the norm not greater than 2BP dim X1 , where BP := supn kPn k. 3. If P = (Pn ) is an approximate identity on X and A ∈ L(X, P) then −2 BP kAkL(X) ≤ kA|X0 kL(X0 ) ≤ kAkL(X) ,

(1.6)

Furthermore, for every T ∈ K(X0 , P) there is a lifting K ∈ K(X, P) of T , that is K|X0 = T . Restrictions and liftings of P-compact projections are again projections of the same rank.

Proof. 1. Let A ∈ L(X, P) and x ∈ X0 . Since kQn Axk ≤ kQn APl kkxk + kQn kkAkkQl xk, where the latter term is smaller than any prescribed > 0 if l is large enough, and the first term tends to zero for any fixed l and n → ∞, we find that Ax ∈ X0 . Hence A|X0 ∈ L(X0 , P). If K ∈ K(X, P) then it is also clear by the definition that K|X0 ∈ K(X0 , P). 2. Recall the projection P := SRPm from the proof of Proposition 1.32 which obviously fulfills kP k ≤ kSkkRkkPm k ≤ 2BP dim im R. 3. For every > 0 there is an m ∈ N such that kAk ≤ kPm Ak + (see Remark 1.18) and for all sufficiently large n we have kPm AQn k ≤ . Now (1.6) easily follows by 2 kA|X0 k ≤ kAk ≤ kPm APn k + 2 ≤ BP kA|X0 k + 2.

Further, let T ∈ K(X0 , P). From kT −Pn T Pn kL(X0 ) → 0 as n → ∞ we conclude for the sequence (Pn T Pn ) ⊂ K(X, P) and n, m large that 2 kPn T Pn − Pm T Pm kL(X) ≤ BP kPn T Pn − Pm T Pm kL(X0 )

2 ≤ BP (kPn T Pn − P kL(X0 ) + kT − Pm T Pm kL(X0 ) ) →n,m→∞ 0.

Hence (Pn T Pn ) ⊂ K(X, P) is a Cauchy sequence. For its limit K ∈ K(X, P) we easily check that the compression K|X0 coincides with T . The rest is easy to prove.

18


Here now comes an extended version of Theorem 1.27. Theorem 1.27*. Let P be a uniform approximate identity. Then A ∈ L(X, P) has the Pdichotomy if one of the following conditions is fulfilled: • P ∗ is an approximate identity on X∗ . • There are a Banach space Y and operators Rn , B ∈ L(Y) such that Y∗ = X, Rn∗ = Pn for all n ∈ N and B ∗ = A. • For every properly P-Fredholm operator B ∈ L(X0 , P) the sequence (BPn ) has a P-strong limit in L(X). • X is complete w.r.t. P-strong convergence. Proof. We prepare the proof with some basic observations in the first and a technical result in the second step. 1st step. Let Pˆ = (Fn ) be a uniform approximate projection given by Theorem 1.15. Notice first that for every fixed k and sufficiently large n, kPk xk = kPk Fn xk ≤ CP kFn xk. Hence, we deduce that lim sup kFn xk ≥ CP−1 kxk for all x ∈ X. (1.7) n→∞

Moreover, for each x ∈ X we have kQn xk → 0 if and only if k(I − Fn )xk → 0 as n → ∞. Further we write m Pˆ n if Fk (I − Fl ) = (I − Fl )Fk = 0 for all k ≤ m and all l ≥ n. 2nd step. Suppose that, for given , δ > 0 and x ∈ X with kxk = 1, we have kAxk ≤ and lim supn k(I − Fn )xk ≥ δ. Then there is an integer N0 such that, for all n, m ≥ N0 , kAFn xk ≤ k[A, Fn ]k + kFn kkAxk ≤ 2CP ,

hence kA(Fn − Fm )xk ≤ 4CP .

Choose n1 Pˆ N0 such that k(I − Fn1 )xk ≥ δ/2 and m1 Pˆ n1 such that, due to (1.7), k(Fm1 − Fn1 )xk = kFm1 (I − Fn1 )xk ≥ (2CP )−1 k(I − Fn1 )xk ≥ (4CP )−1 δ. (F

−F

)x

m1 n1 Set x1 := k(Fm ˆ m1 . −Fn1 )xk and fix N1 P 1 We iterate this procedure as follows: Suppose that nk , mk , Nk , xk for k = 1, . . . , l − 1 are given. Then we analogously choose nl Pˆ Nl−1 such that k(I − Fnl )xk ≥ δ/2, and ml Pˆ nl such that

(F

−F

)x

ml nl k(Fml − Fnl )xk ≥ (4CP )−1 δ, and set xl := k(Fm ˆ ml . By doing this, we −Fnl )xk as well as Nl P l obtain a set {x1 , . . . , xl } ⊂ X and integers N0 , . . . , Nl such that (FNk − FNk−1 )xi = δik xi . For Pl y = i=1 αi xi and every k = 1, . . . , l we find

l

l

X

X 1 1 1

kyk = αi xi ≥ kα x k = |αk | α i xi ≥

(FNk − FNk−1 )

kFNk − FNk−1 k

CP k k CP i=1 i=1

and hence

kAyk ≤

l X i=1

|αi |kAxi k ≤

l X i=1

CP kyk

4CP 16CP3 l 4CP = kyk. δ δ

Let Nl+1 Pˆ Nl and choose a projection R ∈ L(ker(I − FNl+1 )) of norm at most l onto span{x1 , . . . , xl } ⊂ ker(I − FNl ) ⊂ ker(I − FNl+1 ), due to Proposition 1.5. Then P := RFNl is a 4 2 16CP l ˆ projection of rank l and kAP k ≤ . Moreover, P is P-compact, hence P-compact. δ


19

3rd step. If there is an x ∈ ker A such that the norms kQn xk do not tend to zero then the second step yields that for every k and every γ > 0 there is a P-compact projection P of rank k such that kAP k < γ, hence A is properly P-deficient from the right. If, on the other hand, kQn xk → 0 as n → ∞ for all x ∈ ker A, i.e. ker A ⊂ X0 then, due to Proposition 1.33, 2. we have that for Fredholm operators A there is always a P-compact projection onto its kernel, and if A has an infinite dimensional kernel then it is properly Pdeficient. 4th step. Suppose that A|X0 is not normally solvable, that is for every fixed > 0 and every k ∈ N there is a subspace X1 ⊂ X0 of dimension k such that kA|X1 k ≤ /(2kBP ) (see Proposition 1.6). By Proposition 1.33, 2. we can choose a projection P ∈ K(X, P) onto X1 with the norm not greater than 2kBP . Then kAP k ≤ kA|X1 kkP k ≤ . Since and k are arbitrary, we deduce the proper P-deficiency of A from the right. Now suppose that A|X0 is normally solvable and A has a finite dimensional kernel contained in X0 but A is not normally solvable. Let X2 denote a complement of ker A in X. Then the operator A|X2 : X2 → X is still not normally solvable, but the compression A|X3 : X3 → X0 to the space X3 := {x ∈ X2 : kQn xk → 0 as n → ∞} which is a complement of ker A in X0 is normally solvable, hence C := inf{kAxk : x ∈ X3 , kxk = 1} > 0. Then for every x ∈ X2 , kxk = 1 with dist(x, X3 ) < 1/4 min{1, C/kAk} we have kAxk ≥ C/2. Consequently, there is a δ > 0 such that for every > 0 there is an x ∈ X2 with kxk = 1, lim supn k(I − Fn )xk ≥ δ and kAxk ≤ , and the second step again yields the proper P-deficiency from the right. 5th step. It remains to consider operators A which are normally solvable and dim ker A < ∞. Notice that dim coker A = dim ker A∗ in this case, and due to the above considerations A|X0 is normally solvable, too, hence dim coker A|X0 = dim ker(A|X0 )∗ . Obviously, P is a uniform approximate identity on X0 , which tends strongly to the identity, and the sequence P ∗ = (Pn∗ ) still forms a uniform approximate identity on (X0 )∗ . Indeed, for a functional f ∈ (X0 )∗ and for > 0 let x ∈ X0 , kxk = 1 be such that |f (x)| ≥ kf k − . Then kPn∗ f k ≥ |(Pn∗ f )(x)| = |f (Pn x)| ≥ |f (x)| − |f (Qn x)| ≥ kf k − − kf kkQn xk, hence supn kPn∗ f k ≥ kf k. Thus, we can apply the previous steps to (A|X0 )∗ and find, for every k ≤ dim coker A|X0 , a P ∗ -compact rank-k-projection T such that (A|X0 )∗ T = 0. As in the proof of Proposition 1.32 this yields a P-compact projection P 0 of rank k such that P 0 A|X0 = 0. For its lifting P˜ 0 which is given by Proposition 1.33 we find that also P˜ 0 A = 0 by the estimate kP˜ 0 Ak ≤ kP˜ 0 APn k + kP˜ 0 AQn k ≤ kP 0 A|X0 kkPn k + kP˜ 0 AQn k,

(1.8)

where the first term equals zero and the second one tends to zero as n → ∞. Thus, we conclude that if dim coker A|X0 = ∞ then A|X0 and A are properly P-deficient from the left. Moreover, if dim coker A|X0 < ∞ then A|X0 is properly P-Fredholm, and if, additionally, dim coker A = dim coker A|X0

(1.9)

would be true then A would be properly P-Fredholm as well. 6th step. If P ∗ is an approximate identity then the idea of the 5th step can be directly applied to A instead of A|X0 , and the assertion of the theorem is proved for this case. 7th step. If there is a predual space Y and operators (Rn ) as supposed in the second condition of the theorem then notice that (Rn ) is a uniform approximate identity on Y. Indeed, assume that there is a y ∈ Y with supn kRn yk < kyk. Then y ∈ / Y0 . Thus, a bounded linear functional x ∈ X exists with |x(y)| > 0 and x(Y0 ) = {0}, that is x 6= 0 but Pn x = x ◦ Rn∗ = 0 for all n, a contradiction. Moreover, k(Rn − I)Rm k = kPm (Pn − I)k, kRm (Rn − I)k = k(Pn − I)Pm k, and

20


kRU k = kPU k for all bounded U ⊂ R. Thus, the preadjoint operator B has the (Rn )-dichotomy by what we have already proved and the assertion is now also obvious for A. 8th step. We show that the third condition in the theorem implies (1.9). Let dim ker A < ∞ as well as dim coker A|X0 < ∞. Then A|X0 : X0 → X0 is properly P-Fredholm and we can choose a P-regularizer B such that P 0 := I − AB ∈ K(X0 , P) is a projection parallel to the image of A|X0 . ˜ denote the P-strong limit of (BPn )n which is in L(X, P) (see Theorem 1.19). Further, let Let B P˜ 0 denote the lifting of P 0 . With the help of Remark 1.18 we find an m such that ˜ ≤ 2kPm (I − P˜ 0 − AB)k ˜ kI − P˜ 0 − ABk

˜ + 2kPm (I − P˜ 0 )Qn k ≤ 2kPm ((I − P 0 )Pn − AB)k ˜ + 2kPm (I − P˜ 0 )Qn k. = 2kPm A(BPn − B)k

˜ Together with P˜ 0 A = 0 due Since both items tend to zero as n → ∞ we see that P˜ 0 = I − AB. 0 ˜ to (1.8) this yields that P is a projection parallel to im A. Hence A is Fredholm, together with step 3 also P-Fredholm, and the assertion of the theorem is proved as well. 9th step. We finally show that the last condition of the theorem implies the third one. For this, let B ∈ L(X0 , P) be properly P-Fredholm. Then (BPn )n is bounded and (Pk BPn )n is a Cauchy sequence in L(X, P) for every fixed k, since kPk BPn − Pk BPm k ≤ kPk BQl kkPn − Pm k for n, m l, and kPk BQl k → 0 as l → ∞, hence there are uniform limits Bk ∈ L(X, P) of (Pk BPn )n∈N for each k. Moreover, due to our assumption, for each x ∈ X, there is a uniquely determined P-strong limit ˜ := P-lim BPn x in X, Bx n→∞

˜ defines a bounded linear operator B. ˜ Obviously, B| ˜ X = B and by and the mapping x 7→ Bx 0 ˜ − Bk )xk ≤ kPk (B ˜ − BPn )xk + kPk BPn − Bk kkxk → 0 as n → 0, k(Pk B

˜ = Bk . Hence for every x ∈ X, we see that Pk B

˜ − BPn )k = kBk − Pk BPn k → 0 as n → ∞, kPk (B ˜ − BPn )Pk k = k(B ˜ − B)Pk k = 0 for n k. k(B

˜ Thus, (BPn ) converges P-strongly to B. In summary, we see that in the case of uniform approximate identities every operator A ∈ L(X, P) has the P-dichotomy, if X is small (in the sense that P ∗ should be an approximate identity) or large (in the sense of being complete w.r.t. P-strong convergence). Until now it is not clear if there really is a gap between these two extremal cases. The exact formulation of this open question is as follows: Are there a Banach space X, a uniform approximate identity P and an operator A ∈ L(X, P) which is normally solvable, and fulfills dim ker A < ∞ and dim coker A|X0 < dim coker A?

In the Fredholm theory for band-dominated operators on l∞ the existence of a predual setting played an important role to ensure that the Fredholm properties of A and A|X0 coincide.10 To integrate this aspect into the picture which arises from the present considerations we note that, by the previous theorem, a predual setting always implies the P-dichotomy. On the other hand, the P-dichotomy guarantees the required connection between A and A|X0 : 11 10 See, 11 This

for instance, [44] or [15]. particularly clarifies the open problem No. 4 formulated in [15].

1.3. GEOMETRIC CHARACTERISTICS OF BOUNDED LINEAR OPERATORS

21

Proposition 1.34. Let P be a uniform approximate projection on the Banach space X and let A ∈ L(X, P) have the P-dichotomy. Then A is Fredholm if and only if A|X0 is Fredholm. In this case dim ker A = dim ker A|X0 ,

dim coker A = dim coker A|X0 ,

and

ind A = ind A|X0 .

Proof. If A is not Fredholm then it is properly P-deficient, which also yields the P-deficiency of A|X0 ∈ L(X0 , P) (with P regarded as approximate projection on X0 ) by Proposition 1.33. Hence A|X0 can not be Fredholm as well, as shown by the fourth condition in Theorem 1.3. Vice versa, let A be Fredholm and properly P-Fredholm. Corollary 1.31 provides an operator B in L(X, P) such that ABA = A, BAB = B and P = I − BA, P 0 = I − AB ∈ K(X, P). Then P is a projection onto the kernel of A and parallel to the range of B (see Proposition 1.7). Analogously, P 0 is a projection onto the kernel of B and parallel to the range of A. Thus, ker B is a complement of im A, ker A = im P = im P |X0 = ker A|X0 , and ker B = im P 0 = im P 0 |X0 = ker B|X0 . This proves dim ker A = dim ker A|X0 . By Proposition 1.33 we find that the compressions also fulfill A|X0 B|X0 A|X0 = A|X0 , and the projection P 0 |X0 = I − A|X0 B|X0 is onto ker B|X0 and parallel to im A|X0 . Consequently, ker B|X0 is a complement of im A|X0 and we conclude that dim coker A = dim coker A|X0 . The remaining statements now easily follow.

1.3 1.3.1

Geometric characteristics of bounded linear operators One-sided approximation numbers and their relatives

Lower approximation numbers 12 For Banach spaces X, Y and an operator A ∈ L(X, Y) the m-th approximation number from the right srm (A) and the m-th approximation number from the left slm (A) of A are defined as srm (A) := inf{kA − F kL(X,Y) : F ∈ L(X, Y), dim ker F ≥ m},

slm (A) := inf{kA − F kL(X,Y) : F ∈ L(X, Y), dim coker F ≥ m},

(m = 0, 1, 2, ...), respectively. It is clear that 0 = sr0 (A) ≤ sr1 (A) ≤ sr2 (A) ≤ ... and that the same holds true for the sequence (slm (A)). Notice that in the case Y = X and dim X < ∞ the rightand left-sided variants coincide. Lower Bernstein and Mityagin numbers Besides the approximation numbers there are further similar geometric characteristics for bounded linear operators A ∈ L(X, Y) on Banach spaces X, Y. Denote by BX the closed unit ball in X and by j(A) := sup{τ ≥ 0 : kAxk ≥ τ kxk for all x ∈ X},

q(A) := sup{τ ≥ 0 : A(BX ) ⊃ τ BY }

the injection modulus and the surjection modulus, respectively. Due to [54], B.3.8 we have j(A) = inf{kAxk : x ∈ X, kxk = 1},

j(A∗ ) = q(A) and q(A∗ ) = j(A).

(1.10)

Furthermore, for given closed subspaces V ⊂ X and W ⊂ Y we let JV denote the embedding map of V into X and by QW the canonical map of Y onto the quotient Y/W . We define the lower Bernstein and Mityagin numbers by Bm (A) := sup{j(AJV ) : dim X/V < m}, Mm (A) := sup{q(QW A) : dim W < m}. 12 The

content of Section 1.3 was already published, up to some minor modifications, in [81].

22


These characteristics have been discussed in [65], for instance. We will show that there are estimates that connect them to the approximation numbers defined above. Furthermore, these estimates will allow to relate the approximation numbers to the Fredholm property and the norm of the inverse of an operator as well as to the lower singular values in case of X being a Hilbert space. Note also that the sequences (Bm (A)), (Mm (A)) are monotonically increasing and the following auxiliary result holds. Proposition 1.35. Let X, Y be Banach spaces, A ∈ L(X, Y) and R ∈ L(Y) be a projection. Then q(Qker R A) = j(A∗ Jim R∗ ). In particular, Mm (A) = Bm (A∗ ) holds for every m ∈ N. Proof. Define an operator T : im R∗ → (Y/ ker R)∗ by T f = g, g(y + ker R) := f (y). T is well defined and surjective since all f ∈ im R∗ vanish on ker R and since every functional g on Y/ ker R induces a functional g◦Qker R on Y which, due to Qker R = Qker R ◦R+Qker R ◦(I −R) = Qker R ◦R, even fulfills the relation g ◦ Qker R = g ◦ Qker R ◦ R = R∗ (g ◦ Qker R ), that is it belongs to im R∗ . Furthermore, T is an isometry since kT f k =

|(T f )(Qker R (y))| = sup kQker R (y)k y∈Y\ker R y∈Y\ker R sup

|f (y)| inf ky + zk

z∈ker R

|f (y + z)| |f (y)| = sup sup = kf k, = sup sup y∈Y\ker R z∈ker R ky + zk y∈Y\ker R z∈ker R ky + zk and (Q∗ker R (T f )) (y) = (T f )(Qker R y) = f (y) holds for all f ∈ im R∗ and all y ∈ Y. Hence q(Qker R A) = j((Qker R A)∗ ) = inf{kA∗ Q∗ker R gk : g ∈ (Y/ker R)∗ , kgk = 1} = inf{kA∗ Q∗ker R (T f )k : f ∈ im R∗ , kf k = 1}

= inf{kA∗ f k : f ∈ im R∗ , kf k = 1} = j(A∗ Jim R∗ )

follows with (1.10) which gives the first assertion. Now, notice that, by Proposition 1.5, every subspace W ⊂ Y with dim W < m is the kernel of a certain projection R which provides a subspace U := im R∗ of Y∗ . Vice versa, every subspace U ⊂ Y∗ with dim Y∗ /U < m induces a subspace W := {y ∈ Y : f (y) = 0 ∀f ∈ U } ⊂ Y and for a projection S onto W we see that S ∗ f = f ◦ S = 0 for every f ∈ U which easily yields that R := I − S fulfills W = ker R and U = im R∗ . Thus, the coincidence of Mm (A) and Bm (A∗ ) follows from their definitions. Proposition 1.36. Let X, Y be Banach spaces and A ∈ L(X, Y). Then, for all m ∈ N, srm (A) ≤ Bm (A) ≤ srm (A), 2m − 1

as well as

slm (A) ≤ Mm (A) ≤ slm (A). 2m − 1

Proof. Let F ∈ L(X, Y) with dim ker F ≥ m and let V ⊂ X be a subspace with the codimension codim V = dim X/V < m. Then ker F ∩ V is at least 1-dimensional. Thus kA − F k = sup{k(A − F )xk : x ∈ X, kxk = 1} ≥ sup{kAxk : x ∈ ker F, kxk = 1} ≥ inf{kAxk : x ∈ (ker F ∩ V ), kxk = 1} ≥ inf{kAxk : x ∈ V, kxk = 1} = j(AJV ).

Since V was chosen arbitrarily, it follows that kA − F k ≥ Bm (A), and since F is arbitrary, too, we find that srm (A) ≥ Bm (A). Now, let > 0. We inductively construct rank-k-projections Sk (k = 1, . . . , m) and respective elements xk ∈ ker Sk−1 with kxk k = 1 such that ck−1 ≤ ck := kAxk k < j(AJker Sk−1 ) + ,


23

as well as functionals fk ∈ X∗ as follows: Set S0 := 0 and c0 := 0. If the projections S0 , . . . , Sk−1 together with their corresponding elements and functionals are already given, we choose xk in ker Sk−1 and a functional f˜k on ker Sk−1 with kf˜k k = 1 and f˜k (xk ) = 1. Furthermore, we define Pk the functional fk := f˜k ◦ (I − Sk−1 ) and the projection Sk x := i=1 fi (x)xi on X. Notice that we obviously have fi (xj ) = δij 13 for all i, j ≤ k and from kfk k ≤ kI − Sk−1 k ≤ 1 +

k−1 X i=1

kfi k

it follows that kfk k ≤ 2k−1 , hence

k

k k

X

X X

kASk xk = fi (x)Axi ≤ ci kfi kkxk ≤ ck 2i−1 kxk = ck (2k − 1)kxk.

i=1

i=1

i=1

Defining V := ker Sm−1 , we find

srm (A) = inf{kA − F k : dim ker F ≥ m} ≤ kA − A(I − Sm )k

= kASm k ≤ cm (2m − 1) < (2m − 1)(j(AJV ) + ) ≤ (2m − 1)(Bm (A) + ),

(1.11)

which completes the proof of the first estimate, since is arbitrary. From Proposition 1.35 and the above applied to A∗ we know that Mm (A) = Bm (A∗ ) ≤ srm (A∗ ). Fix > 0 and an operator F ∈ L(X, Y) with dim coker F ≥ m such that kA − F k ≤ slm (A) + . Choose a rank m projection Q such that kQF k < (see Proposition 1.6). Then the kernel of F ∗ (I − Q)∗ is at least m-dimensional and srm (A∗ ) ≤ kA∗ − F ∗ (I − Q)∗ k ≤ kA∗ − F ∗ k + kF ∗ Q∗ k = kA − F k + kQF k ≤ slm (A) + 2 gives the asserted estimate Mm (A) ≤ slm (A) since was chosen arbitrarily. For the remaining part of the second estimate in the assertion we use a construction similar to the one above. Fix > 0 and set R0 := 0, d0 := 0. For k = 1, . . . , m we gradually choose ∗ functionals gk ∈ ker Rk−1 with kgk k = 1 such that ∗ dk−1 ≤ dk := kA∗ gk k < j(A∗ Jker Rk−1 ) + ,

as well as elements y˜k ∈ Y with k˜ yk k = 1 and |gk (˜ yk )| ≥ 1 − , respectively. Furthermore, we Pk define yk := (gk (˜ yk ))−1 (I − Rk−1 )˜ yk and an operator Rk y := i=1 gi (y)yi on Y each time. We easily check that gi (yj ) = δij for all i, j ≤ k hence Rk is a projection of rank k, and from ! k−1 X 1 1 kI − Rk−1 k ≤ 1+ kyi k kyk k ≤ 1− 1− i=1 we conclude kyk k ≤

2k−1 . (1−)k

Thus, for all g ∈ Y∗ and all x ∈ X,

k

k

!

X

X

k(A∗ Rk∗ g)(x)k = kg(Rk Ax)k = gi (Ax)g(yi ) = g(yi )A∗ gi (x)

i=1

k X

i=1

k X 2i−1 2k − 1 ≤ kgkkyi kdi kxk ≤ dk kgkkxk ≤ dk kgkkxk. i (1 − ) (1 − )k i=1 i=1

13 δ

ij

= 1 if i = j, and δij = 0 otherwise

24


The assertion then follows by ∗ slm (A) = inf{kA − F k : dim coker F ≥ m} ≤ kRm Ak = kA∗ Rm k ≤ dm

≤

2m − 1 2m − 1 ∗ (B (A ) + ) = (Mm (A) + ) m (1 − )m (1 − )m

2m − 1 (1 − )m

where > 0 is arbitrary. Now we have sr1 (A) = B1 (A) = j(A) and sl1 (A) = M1 (A) = q(A) = j(A∗ ), hence we deduce Corollary 1.37. Let A ∈ L(X, Y). Then ( kA−1 k−1 if A is invertible r/l s1 (A) = 0 if A is not invertible from the left/right. With the help of Proposition 1.6 we find even more. Corollary 1.38. Let A ∈ L(X, Y). 1. If A is normally solvable and k is greater than the dimension of the kernel (cokernel) of A then srk (A) (slk (A), resp.) is non-zero. Otherwise srk (A) (slk (A)) is equal to zero. 2. If A is not normally solvable then all approximation numbers are equal to zero. In particular, A is Fredholm if and only if the number of vanishing approximation numbers of A is finite.

1.3.2

Hilbert spaces and lower singular values

Suppose now that X, Y are Hilbert spaces and A ∈ L(X, Y). We denote by A? the Hilbert adjoint ? as introduced in Section 1.1.1 and consider the essential spectrum of the operator pA A which is 14 a subset of R+ , the set of all non-negative real numbers. Furthermore, let d := inf σess (A? A) and σ1 (A) ≤ σ2 (A) ≤ . . .

be the sequence of the non-negative square roots of the eigenvalues of A? A less than d, counted according to their multiplicities. If there are only N A (= 0, 1, 2, . . .) such eigenvalues, we put σN A +1 (A) = σN A +2 (A) = . . . = d. These numbers may be called the lower singular values of A. Corollary 1.39. Let X, Y be Hilbert spaces and A ∈ L(X, Y). Then, for all m, srm (A) = Bm (A) = σm (A),

as well as

slm (A) = Mm (A) = σm (A? ).

Proof. Firstly, suppose that A is normally solvable and has a trivial kernel. Set B := A? A and for a subspace U ⊂ X let U ⊥ denote its orthogonal complement. Then (Bm (A))2 = sup{inf{kAxk2 : x ∈ U ⊥ , kxk = 1} : dim U < m}

= sup{inf{hx, Bxi : x ∈ U ⊥ , kxk = 1} : dim U < m}

which equals (σm (A))2 by the Min-Max Principle (see [66], Theorem XIII.1 or below). 14 See

the definition in Section 1.4 and Theorem 1.40, and note that (A? A)? = A? A.


25

If A is not normally solvable or dim ker A = ∞ then srm (A) = Bm (A) = 0 for all m, due to Corollary 1.38. Moreover, B is not Fredholm in this case, that is 0 ∈ σess (B) hence σm (A) = 0 for all m. In the case that A is normally solvable and k = dim ker A < ∞ we set Aˆ := AJ(ker A)⊥ and find that ˆ = σm (A) ˆ = σm+k (A) Bm+k (A) = Bm (A) for all m, as well as Bl (A) = σl (A) = 0 for all l ≤ k. Moreover, for a subspace U of dimension m and PU the orthogonal projection onto U , (srm (A))2 ≤ kAPU k2 = sup{hAx, Axi : x ∈ U, kxk ≤ 1}

≤ sup{kxkkBxk : x ∈ U, kxk ≤ 1} ≤ kBJU k.

(1.12)

2 If m ≤ N A , then there is an orthonormal system {xi }m i=1 of eigenvectors, i.e. Bxi = (σi (A)) xi . m Set U := span{xi }i=1 and deduce from (1.12) that (m ) m X X (srm (A))2 ≤ sup αi2 (σi (A))2 : x = αk xk , kxk ≤ 1 ≤ (σm (A))2 . i=1

A

k=1

2

For the case m > N we note that C := B−d I is not Fredholm, since d2 ∈ σess (B). That is, C is not normally solvable or dim ker C = dim coker C = ∞. Thus, by Proposition 1.6, for every > 0 there is a subspace U of dimension m such that kCJU k < , hence (srm (A))2 ≤ kBJU k < d2 + . Together with Proposition 1.36 this finishes the proof of the first assertion, and the second one follows by duality, Proposition 1.35 and since slm (A) = srm (A? ) in the case of Hilbert spaces. For completeness we state the Min-Max-Principle from [66], Theorem XIII.1 here, where we call H ∈ L(X) self-adjoint, if H = H ? , that means hHx, yi = hx, Hyi for all x, y ∈ X. Theorem 1.40. Let H ∈ L(X) be a normally solvable, injective, and self-adjoint operator on the Hilbert space X and define µn (H) := supU inf{hx, Hxi : x ∈ U ⊥ , kxk = 1}, the supremum over all subspaces U of dimension at most n − 1. Then, for each fixed n, exactly one of the following hold: • There are n eigenvalues (counted according to their algebraic multiplicities) below the bottom of the essential spectrum, and µn (H) is the nth eigenvalue. • µn (H) is the bottom of the essential spectrum. Remark 1.41. Besides the lower approximation, Bernstein and Mityagin numbers there are also their much more famous “upper relatives”. For example, the upper approximation numbers sk (A) of an operator A are given by sk (A) := inf{kA − F kL(X,Y) : F ∈ L(X, Y), rank F < k}. These numbers form a decreasing sequence kAk = s0 (A) ≥ s1 (A) ≥ . . . ≥ 0 and, roughly speaking, they sound out the spectrum of an operator from above, and not from below as the lower approximation numbers do. For the other characteristics the definitions are similar. Notice that all of these “big brothers” are so-called s-numbers, for which a well developed theory is available that includes many results on the relations between them. The modern books [53] and [22] may provide a good overview and introduction to that business. Unfortunately, as far as we know, there are no results in the literature for the lower versions in the case of (infinite dimensional) Banach spaces, except those of [65], from where we borrowed the equalities Bm (A) = σm (A) and Mm (A) = σm (A? ). In particular, note that the estimates in Proposition 1.36 seem hardly to be optimal. Nevertheless, they suffice for the aims of the present text.

26

1.4 1.4.1


Spectral properties The spectrum

Let A be a Banach algebra with identity I and A ∈ A. The spectrum sp A of A is the set of all complex numbers z ∈ C such that A − zI is not invertible in A. From any textbook on functional analysis or spectral theory it is well known that the spectrum is a non-empty compact set. Hence, the spectral radius ρ(A) of A ρ(A) := sup{|z| : z ∈ sp A} is well defined and the Spectral Radius Formula states that ρ(A) = inf kAn k1/n = lim kAn k1/n . n∈N

n→∞

We particularly get the spectrum of a bounded linear operator A ∈ L(X) on a Banach space X: sp A := {z ∈ C : A − zI is not invertible}.

Further, the essential spectrum spess A of A is defined as the spectrum of A + K(X) in the Calkin algebra L(X)/K(X) or, equivalently due to Theorem 1.3, spess A := {z ∈ C : A − zI is not Fredholm}.

1.4.2

The (N, )-pseudospectrum

In [30] the authors write “A computer working with finite accuracy cannot distinguish between a non-invertible matrix and an invertible matrix the inverse of which has a very large norm”. Therefore one replaces the spectrum by so-called pseudospectra which reflect finite accuracy. For some pioneering work we refer to Landau [40], [41], Reichel and Trefethen [67] Böttcher [6], and for a (or more striking: the) comprehensive source we recommend the monograph [87] of Trefethen and Embree. Definition 1.42. For N ∈ Z+ and > 0 the (N, )-pseudospectrum of a bounded linear operator A on a Banach space X is defined as the set N

−N

spN, A := {z ∈ C : k(A − zI)−2 k2

≥ 1/}.15

Remark 1.43. Notice that (for N = 0) this definition of the (N, )-pseudospectrum includes the definition of the (classical) -pseudospectrum sp A := {z ∈ C : k(A − zI)−1 k ≥ 1/}. The -pseudospectra have gained attention after [67] and [6] disclosed that, on the one hand they approximate the spectrum but are less sensitive to perturbations, and on the other hand the -pseudospectra of discrete convolution operators mimic exactly the -pseudospectrum of an appropriate limiting operator, which is in general not true for the “usual” spectrum. See also [4], [13], [30], [11] and the references cited there. Later on, Hansen [32], [33] introduced the (N, )-pseudospectra for linear operators on separable Hilbert spaces and pointed out that they share several nice properties with case N = 0, but offer a better insight into the approximation of the spectrum. Furthermore, it was shown how the spectrum can be approximated numerically, based on the consideration of singular values of certain finite matrices. We now recover and extend these results in what follows and in Section 3.2.3. 15 Here

we use the convention kB −1 k = ∞ if B is not invertible.

1.4. SPECTRAL PROPERTIES

27

Within this section we want to show that the (N, )-pseudospectra provide a nice tool for the approximation of the spectrum of a bounded linear operator A, even in the Banach space case. More precisely, we will show that Theorem 1.44. Let A ∈ L(X). For every δ > > 0 there is an N0 such that, for all N ≥ N0 , B (sp A) ⊂ spN, A ⊂ Bδ (sp A),

(1.13)

where B (S) := {z ∈ C : dist(z, S) ≤ } denotes the closed -neighborhood of the set S. In a very recent preprint [34] Hansen and Nevanlinna mentioned that this result is in force even in the Banach space context, but the Hilbert space approach for the approximate determination of the spectrum via singular values of finite matrices cannot be extended to the Banach case since there is no involution available anymore. Therefore we propose a modification of Hansens idea which replaces the singular values by the injection and surjection modulus. Here the precise description comes: If there is a sequence (Ln ) of compact projections which converge ∗-strongly to the identity then we will obtain an approximation of the spectrum of A based on much simpler operators: For this define the functions 2−N N N m,n γN (z) := min{j((Ln (A − zI)Ln )2 Jim Lm ), q(Qker Lm (Ln (A − zI)Ln )2 )} (1.14) where the operators J, Q are defined as in Section 1.3.1. We are interested in their sublevel sets m,n Γm,n N, := {z ∈ C : γN (z) ≤ }

because for every fixed β > > α > 0 and sufficiently large N, m, n we will get Bα (sp A) ⊂ Γm,n N, ⊂ Bβ (sp A).

(1.15)

Notice that the projections Ln are of finite rank, thus the operators which have to be considered m,n for the evaluation of γN (z) are operators acting on finite dimensional spaces. For the Hilbert space case this particularly means that we only have to consider the smallest singular values of certain finite matrices in order to determine Γm,n N, (by Corollary 1.39). The paper [31] contains an extensive discussion of this case including explicit algorithms and examples. So, we omit a detailed repetition here and focus on the theoretical treatment of the Banach space case instead. Remark 1.45. Actually, we prove the latter for the case that X possesses a uniform approximate identity P with BP = 1 such that X has the P-dichotomy, and the sequence (Ln ) of P-compact projections converges P-strongly to the identity. In such a setting the asserted relation (1.15) is true for A ∈ L(X, P). Anyway, the proofs work in both cases. In [78] the author already published these results, proofs and discussions of Section 1.4 for the “classical” setting without P and under the additional condition kLn k = 1 for all n. A level function for the (N, )-pseudospectrum Define ( N −N k(A − zI)−2 k−2 if z ∈ / sp A , and γ(z) := dist(z, sp A). γN (z) := 0 if z ∈ sp A The function γ is continuous everywhere, equals zero in all points z ∈ sp A, and γ(z) =

1 1 = N −N = lim γN (z) N →∞ ρ((A − zI)−1 ) lim k(A − zI)−2 k2 N →∞

28


for every z ∈ / sp A. The last equation is the definition of γN (z), the second one is given by the Spectral Radius Formula, and for the first one consider 1 −1 −1 −1 (A − zI) − yI = (A − zI) [I − y(A − zI)] = (A − zI) y z + I −A . y We see that (A − zI)−1 − yI is not invertible if and only if z + o n 1 : x ∈ sp A , and thus that is sp((A − zI)−1 ) = x−z ρ((A − zI)

−1

1 y

belongs to the spectrum of A,

1 1 1 : x ∈ sp A = = . ) = sup x − z inf {|x − z| : x ∈ sp A} dist(z, sp A)

The γN are continuous in every z ∈ / sp A and (pointwise) monotonically increasing w.r.t. N since N +1

k(A − zI)−2

−(N +1)

k2

N

≤ (k(A − zI)−2 k2 )2

−(N +1)

N

−N

= k(A − zI)−2 k2

.

(1.16)

Combining these observations we see that 0 ≤ γN (z) ≤ γ(z), the functions γN are continuous everywhere, and γN (z) converges increasingly to γ(z) for every z ∈ C. By Dini’s Theorem this even gives uniform convergence on every compact subset of C. Fix δ > > 0. It is clear from the definition that z ∈ spN, A if and only if γN (z) ≤ . Choose r > 0 large enough to guarantee that γN (z) > for all z ∈ C\Ur (0) and all N . This is possible by a Neumann series argument since A is bounded. Then we have uniform increasing convergence of γN (z) to γ(z) on Br (0). Thus, there is an N0 such that for every N ≥ N0 and every z ∈ C γ(z) ≤ ⇒ γN (z) ≤ ⇒ γ(z) ≤ δ, which yields (1.13) and finishes the proof of Theorem 1.44. Uniform approximations for γN (z) Notice that by Corollary 1.37 2−N N N γN (z) = min{j((A − zI)2 ), q((A − zI)2 )} . We now use this representation as a starting point for the definition of approximating substitutes: Let P be a uniform approximate identity on the Banach space X with BP = 1 and suppose that X has the P-dichotomy. Further suppose that (Ln ) is a sequence of P-compact projections which converge P-strongly to the identity. For A ∈ L(X, P) set 2−N N N m γN (z) := min{j((A − zI)2 Jim Lm ), q(Qker Lm (A − zI)2 )} .

m Proposition 1.46. For every N and m, the functions γN (z) are continuous w.r.t. z ∈ C. m Further, for every fixed point z ∈ C, the sequence (γN (z))m is bounded below by γN (z) and converges to γN (z). The convergence is even uniform on every compact subset of C.

We first state an auxiliary result. Proposition 1.47. Let X, Y, Z be Banach spaces and A ∈ L(X, Y) as well as B ∈ L(Y, Z). Then j(BA) ≥ j(B) j(A) and q(BA) ≥ q(B) q(A).


29

Proof. If one of the numbers j(A), j(B), q(A), q(B) equals zero then the respective assertion is obviously true. So, let these four numbers be strictly positive and check that kBAxk j(BA) = inf{kBAxk : x ∈ X, kxk = 1} = inf kAxk : x ∈ X, kxk = 1 kAxk ≥ inf{kByk : y ∈ Y, kyk = 1} inf{kAxk : x ∈ X, kxk = 1} = j(B) j(A).

Further, fix σ < q(B) and τ < q(A). Then σBZ ⊂ B(BY ) and τ BY ⊂ A(BX ). Consequently, στ BZ ⊂ τ B(BY ) = B(τ BY ) ⊂ B(A(BX )) = BA(BX ) which shows that στ ≤ q(BA) and finishes the proof. Proof of Proposition 1.46. The continuity is obvious by the relations (1.10), and we immediately conclude the estimate j(BJim Lm ) ≥ j(B)j(Jim Lm ) = j(B) for every m from the previous result. With Proposition 1.35 also q(Qker Lm B) ≥ q(Qker Lm )q(B) = j(Jim L∗m )q(B) = q(B) holds. These N observations particularly hold for all B := (A − zI)2 and hence, in every z, we get the lower m (z))m . bound γN (z) for (γN If B ∈ L(X, P) is invertible then, due to Theorem 1.14, B −1 belongs to L(X, P) and we have j(B) = q(B) = kB −1 k−1 = kB −1 |X0 k−1 by Proposition 1.33, since BP = 1. Given δ > 0 choose x ∈ X0 , kxk = 1, such that kB −1 |X0 k−1 > kB −1 xk−1 − δ and set y := B −1 x, w := kyk−1 y. Then j(B) = q(B) = kB −1 |X0 k−1 >

kByk kxk −δ = − δ = kBwk − δ. −1 kB xk kyk

The latter can be further estimated by kBwk ≥ kBLm wk − kB(I − Lm )wk

kBLm wk 1 − kLm wk − kBLm wk − kB(I − Lm )wk kLm wk kLm wk k(I − Lm )wk ≥ j(BJim Lm ) − kBLm k − kB(I − Lm )wk 1 − k(I − Lm )wk =

(1.17)

where k(I − Lm )wk → 0 as m → ∞ since w ∈ X0 and (Lm ) converges P-strongly to the identity. Also note that (Lm ) is bounded, due to Theorem 1.19. Since δ was chosen arbitrarily, we find that j(BJim Lm ) → j(B) as m → ∞. Plugging this in the estimate

2−N 2−N N N m j((A − zI)2 Jim Lm ) ≥ γN (z) ≥ γN (z) = j((A − zI)2 ) ,

z∈ / sp A

m we deduce that γN (z) converges to γN (z) for every z ∈ / sp A. Now, let B be P-deficient from the right, or let the kernel of B be non-trivial. Fix δ > 0 and a P-compact rank-one-projection P such that kBP k < δ. Let w ∈ im P ⊂ X0 , kwk = 1 and apply (1.17) to deduce

δ > kBwk ≥ j(BJim Lm ) −

k(I − Lm )wk kBLm k − kB(I − Lm )wk, 1 − k(I − Lm )wk

which yields j(BJim Lm ) → 0 as m → ∞ also in this case, since δ > 0 was chosen arbitrarily.

30


If B is P-deficient from the left, or the cokernel of B is non-trivial then, for given δ > 0, we choose a P-compact rank-one-projection P such that kP Bk < δ as well as a functional f ∈ im P ∗ , kf k = 1, and find again by (1.17) and Proposition 1.35 that k(I − L∗m )f k kB ∗ L∗m k − kB ∗ (I − L∗m )f k 1 − k(I − L∗m )f k k(I − L∗m )P ∗ k ≥ q(Qker Lm B) − kB ∗ L∗m k − kB ∗ kk(I − L∗m )P ∗ k. 1 − k(I − L∗m )P ∗ k

δ > kB ∗ f k ≥ j(B ∗ Jim L∗m ) −

The P-strong convergence of (Lm ) together with the P-compactness of P prove q(Qker Lm B) → 0 as m → ∞ in that case, too. Since X has the P-dichotomy, there are no further possible cases, m hence we get pointwise convergence of the continuous functions γN (z) to the continuous function γN (z) in the whole complex plane. m Let M ⊂ C be a compact set. It remains to show the uniform convergence of γN to γN on m M . For this we construct a sequence (fN )m of continuous functions defined on M such that m m m fN (z) ≥ γN (z) ≥ γN (z) and fN (z) converge decreasingly to γN (z) for every z ∈ M . Then m Dini’s Theorem applies to (fN ), provides its uniform convergence, and hence also the uniform m m convergence of (γN ). So, let us construct the desired functions (fN ). Let k ∈ N. For each N 2 B ∈ R := {(A−zI) : z ∈ M } there is a w ∈ im Lk , kwk = 1 such that, using the estimate (1.17) again, j(BJim Lk ) = inf{kBxk : x ∈ im Lk , kxk = 1} ≥ kBwk −

1 2k

1 k(I − Lm )wk kBLm k − kB(I − Lm )wk − 1 − k(I − Lm )wk 2k k(I − Lm )Lk k 1 ≥ j(BJim Lm ) − kBLm k − kBkk(I − Lm )Lk k − . 1 − k(I − Lm )Lk k 2k

≥ j(BJim Lm ) −

Recall that the operator Lk is P-compact and the sequence (Lm ) converges P-strongly to the identity. Hence, there is an mk ∈ N such that, for every operator B in the bounded set R and every m ≥ mk , the estimate j(BJim LK ) ≥ j(BJim Lm ) − 1/k holds. Without loss of generality we can choose these mk such that mk ≥ 2k for every k, and we conclude j(BJim Lk ) +

2 2 ≥ j(BJim Lm ) + k m

for all k ∈ N, m ≥ mk and B ∈ R.

The same arguments applied to j(B ∗ Jim L∗k ), with Proposition 1.35, yield numbers m ˜ k s.t. q(Qker Lk B) +

2 2 ≥ q(Qker Lm B) + k m

for all k ∈ N, m ≥ m ˜ k and B ∈ R.

Consequently, there is a strictly increasing sequence (ln ) ⊂ N such that min{j(BJim Lln ), q(Qker Lln B)} +

2 2 ≥ min{j(BJim Ls ), q(Qker Ls B)} + ln s

m for all n ∈ N, B ∈ R and s ≥ ln+1 . For ln+1 ≤ m < ln+2 we now define the functions fN by m fN (z)

2−N 2 2N 2N := min{j((A − zI) Jim Lln ), q(Qker Lln (A − zI) )} + ln

and straightforwardly check that they meet all requirements: They can be written in the form −N ln 2N m fN = ((γN ) + l2n )2 , hence they are continuous on M and converge pointwise to γN by what


31

we have already proved. Due to the special choice of the (ln ), this convergence is monotonically m m decreasing and the functions even fulfill fN (z) ≥ γN (z) on M . Until now, we have seen that the function γ can be approximated in a sense by the functions γN , m and further the functions γN are approximations of γN . As a third step we finally approximate m,n m γN by the functions γN given in (1.14): 2−N N N m,n γN (z) = min{j((Ln (A − zI)Ln )2 Jim Lm ), q(Qker Lm (Ln (A − zI)Ln )2 )} . N

For A ∈ L(X, P) it is clear that the sequence ((Ln (A − zI)Ln )2 )n∈N converges P-strongly, hence the arguments of j(·) and q(·) converge in the norm as n → ∞ for every z since Lm is P-compact and Qker Lm = Qker Lm ◦ Lm holds. This convergence is even uniform with respect to z on every compact set, since these arguments are polynomials in z whose (operator-valued) m,n m coefficients converge in the norm. This provides the uniform convergence γN (z) → γN (z) as n → ∞ on every compact subset of C. Regard the compact set M := B2kAk+4β (0). For given β > > α > 0 we choose N0 such that γ(z) − γN (z) > (β − )/2 for all N ≥ N0 and z ∈ M . Furthermore, for N ≥ N0 , we choose m (z) − γN (z) < ( − α)/2 for all z ∈ M and all m ≥ m0 (N ). Finally, we m0 (N ) such that γN take n0 (N, m) large enough to guarantee that for all n ≥ n0 (N, m) and z ∈ M the estimate m,n m (z) − γN (z)| < min{( − α), (β − )}/2 and, additionally, j(Ln Jim Lm ), q(Qker Lm Ln ) ≥ 1/2 |γN hold. The latter is again possible since (Ln ) converges P-strongly to the identity and Lm is P-compact. Then, for every z ∈ M , m γ(z) ≤ α ⇒ γN (z) ≤ α ⇒ γN (z) ≤ α +

−α −α m,n ⇒ γN (z) ≤ α + 2 = 2 2

and m,n m γN (z) ≤ ⇒ γN (z) ≤ +

β− β− β− ⇒ γN (z) ≤ + ⇒ γ(z) ≤ + 2 = β. 2 2 2

Applying Proposition 1.47 we get for arbitrary bounded linear operators B that j(BLn Jim Lm ) = j((BLn )(Ln Jim Lm )) ≥ j(BJim Ln )j(Ln Jim Lm ) ≥ j(BJim Ln )/2 and analogously q(Qker Lm Ln B) ≥ q(Qker Ln B)/2. Thus, it is immediate from the definitions and an estimate similar to (1.16) that, for z ∈ / M, m,n n,n 2γN (z) ≥ γN (z) ≥ γ0n,n (z) = |z|k(Ln − z −1 Ln ALn )−1 k−1 L(im Ln ) > 2β.

The estimate γ(z) > β for z ∈ / M is also obvious. Consequently, for suitably chosen N, m and n, we arrive at (1.15). Let us condense this outcome in a theorem. Theorem 1.48. Let P be a uniform approximate identity on the Banach space X providing X with the P-dichotomy, BP = 1, (Ln ) be a sequence of P-compact projections tending P-strongly to the identity, and let A ∈ L(X, P). For every fixed β > > α > 0 and all sufficiently large N ≥ N0 , m ≥ m0 (N ) 16 and n ≥ n0 (N, m) the set n o 2N 2N 2N Γm,n = z ∈ C : min{j((L (A − zI)L ) J ), q(Q (L (A − zI)L ) )} ≤ n n im L ker L n n m m N,

fulfills

Bα (sp A) ⊂ Γm,n N, ⊂ Bβ (sp A). 16 m

0 (N )

means that m0 depends on N .

32


Notice that, if X is a separable Hilbert space and the Ln are compact, orthogonal projections which are nested in the sense that Ln Ln+1 = Ln+1 Ln = Ln for all n ∈ N and converge strongly to the identity,17 then P = (Ln ) satisfies the conditions of this section. Corollary 1.39 further yields n o N N 2N Γm,n Jim Lm ), σ1 ((Ln (A? − z¯I)Ln )2 Jim Lm )} ≤ 2 . N, = z ∈ C : min{σ1 ((Ln (A − zI)Ln ) We again note that the classical Hilbert space setting and the idea of approximating the spectrum via the consideration of certain “rectangular matrices” and their smallest singular value is subject of [31].

1.4.3

Convergence of sets

Definition 1.49. The Hausdorff distance of two compact sets S, T ⊂ C is defined as dH (S, T ) := max max dist(s, T ), max dist(t, S) . s∈S

t∈T

This function dH forms actually a metric on the set of all non-empty compact subsets of C, and offers an alternative view on the present results. Remember that the sets sp A, B (sp A), spN, A and Γn,m N, are compact. Corollary 1.50. Let A ∈ L(X). Then lim dH (sp A, B (sp A)) = 0

→0

and

lim dH (B (sp A), spN, (A)) = 0.

N →∞

Thus, the (N, )-pseudospectra converge to the spectrum of A with respect to the Hausdorff distance if N → ∞ and → 0. Moreover, lim lim sup lim sup dH (B (sp A), Γm,n N, ) = 0,

N →∞ m→∞

n→∞

that is, the spectrum of A can be approximated w.r.t. the Hausdorff metric even by the sets Γm,n N, . Proof. Firstly, note that for compact sets S, T, U with S ⊂ T ⊂ U we have dH (S, T ) = max dist(t, S) ≤ max dist(u, S) = dH (S, U ). t∈T

u∈U

Clearly, dH (sp A, B (sp A)) ≤ which proves the first assertion. For the second one apply Theorem 1.44 and the estimate dH (B (sp A), spN, (A)) ≤ dH (B (sp A), Bδ (sp A)) ≤ δ − , which holds for every fixed δ > > 0 and sufficiently large N . For the last assertion we employ Theorem 1.48 in the same way. Remark 1.51. We want to point out that the -pseudospectrum does not behave continuously with respect to the parameter in general, that is the inclusion clos{z ∈ C : k(A − zI)−1 k > 1/} ⊂ sp A 17 The

sequence (Ln ) is called a filtration in that case.


33

can be proper. Shargorodsky addressed a paper [84] to this fact, in which an explanation and appropriate examples are given in great detail. The point is that k(A − zI)−1 k, the resolvent norm, can take constant values on open sets. Actually, the history of the investigation of this phenomenon is much longer. Globevnik [26] posed the question: “Can k(λe − a)−1 k be constant on an open subset of the resolvent set?” for an element a of a complex Banach algebra. He could only derive a partial answer, but he was able to show that the answer is No for the resolvent norm of a bounded linear operator on a complex uniformly convex Banach space.18 Unfortunately, it remained rather unnoticed, and so this question emerged again in the 90’s where, independently from the earlier, Böttcher asked this and, together with Daniluk, he tackled the case of bounded linear operators on Hilbert spaces in [6], Proposition 6.1 (see also [18]), and later on the Lp -spaces with 1 < p < ∞ as well. Shargorodsky combined all these observations and completed the picture in [84], where he pointed out that Hilbert spaces and the Lp -spaces are complex uniformly convex by [17], and that the outcome even extends to Banach spaces which have a complex uniformly convex dual space. This particularly permits to cover all Lp -spaces with p ∈ [1, ∞]. Of course, the phenomenon of jumping pseudospectra (if it exists in the underlying setting) is reflected in the behavior of the (N, )-pseudospectra as well but, as Theorem 1.44 and Corollary 1.50 show, it gets less significant in any case, since the difference of the respective sets N

−N

clos{z ∈ C : k(A − zI)−2 k2

> 1/} ⊂ spN, A

becomes small with growing N .

1.4.4

Quasi-diagonal operators

The above observation that we can approximate the spectrum with the help of “rectangular finite sections” raises hope that this may even be possible by the usual finite sections. In fact, this is not true as a simple and well known example demonstrates: Let V denote the shift operator (xi )i∈Z 7→ (xi+1 )i∈Z on lp . It is invertible, has norm 1 and its inverse has norm 1 as well. Hence V − αI is invertible whenever |α| = 6 1 by a Neumann series argument. On the other hand, the spectrum of every finite section equals {0} and also their (N, )-pseudospectra contain 0, whereas the (N, )-pseudospectra of V approximate the unit circle. Actually, there is an explanation and a solution to this dilemma, and we will take up this matter again, after we have the required tools of the Parts 2 and 3 available. For the moment we turn our attention to a class of operators which accomplish our desire. Definition 1.52. Let (Ln ) be a sequence of P-compact projections tending P-strongly to the identity, where P = (Pn ) is a uniform approximate identity on X, which equips X with the P-dichotomy and fulfills BP = 1. An operator A ∈ L(X, P) is called quasi-diagonal with respect to (Ln ) (or (Ln ) is said to quasi-diagonalize A) if k[A, Ln ]k := kALn − Ln Ak → 0 as n → ∞. Note that by a result of Berg [3] every normal operator A on a separable Hilbert space, hence also every self-adjoint operator, has a sequence of orthogonal projections which quasi-diagonalizes A. We will pay some more attention to such operators and discuss further details in Section 4.1. At this stage we restrict our considerations solely to their pseudospectra. 18 For

a definition see Section 3.2.3 where we will have a closer look at this notion.

34


Theorem 1.53. Let A be quasi-diagonal with respect to (Ln ). Then 0 = lim dH (sp A, B (sp A)) →0

= lim lim dH (sp A, spN, A) →0 N →∞

= lim lim lim sup dH (sp A, spN, (Lm ALm )). →0 N →∞ m→∞

Proof. The first and second equation are done by Corollary 1.50. For the last one note that m,m m Γm,m already converges uniformly to zero on every N, = spN, (Lm ALm ) and that γN − γN compact subset of C as m → ∞ since A is quasi-diagonal. We mention that Brown [14] already proved the convergence of the classical -pseudospectra in the Hilbert case, based on the resultsT of the C ∗ -algebra approach of Hagen, Roch and Silbermann [30]. For those we have the relation >0 sp A = sp A to the usual spectrum. Also Hansen [32] considered the spectral approximation of self-adjoint operators, taking their quasi-diagonality as well as their (N, )-pseudospectra into account.

1.5

Example: operators on lp -spaces

Let us summarize what we have already found for the spaces lp = lp (Z, X) and the sequence P = (Pn ) of projections given by Pm : (xi ) 7→ (. . . , 0, x−m , . . . , xm , 0, . . .). For every 1 ≤ p ≤ ∞ and every Banach space X, P forms a uniform approximate identity and lp has the P-dichotomy (cf. Section 1.2.3 and Corollary 1.28). Let A ∈ L(lp , P) and suppose that the finite section sequence (Pn APn ) is stable. Then, for every b ∈ lp , the equation Ax = b has a unique solution x ∈ lp , and also the equations Pn APn xn = Pn b have unique solutions xn (for sufficiently large n), which converge P-strongly to x (Corollary 1.23 and Proposition 1.29). So, all we are left with is the question how to prove the stability of the finite section sequence. We want to discuss this for a more specific class of operators.

1.5.1

Band-dominated operators

Every sequence a = (ai ) ∈ l∞ (Z, L(X)) gives rise to an operator aI ∈ L(lp ) (a so-called multiplication operator) via (ax)i = ai xi , i ∈ Z. Evidently, kaIkL(lp ) = kak∞ . By this means, the sequences in l∞ (Z, C) induce multiplication operators as well. P Definition 1.54. A band operator A is a finite sum of the form A = α aα Vα , where α ∈ Z, aα are multiplication operators, and Vα denote the shift operators (Vα f )(x) := f (x − α),

x ∈ Z, f ∈ lp .

An operator is called band-dominated if it is the uniform limit of a sequence of band operators. We denote the class of all band-dominated operators by Alp .

1.5. EXAMPLE: OPERATORS ON LP -SPACES

35

Clearly, in case X = C, the matrix representation of a multiplication operator aI with respect to the canonical basis in lp is an infinite diagonal matrix with the numbers ai along the main diagonal. The matrix representations of band operators are matrices having a finite bandwidth, that is there are only finitely many non-zero (sub- or super-)diagonals. This is why we call the uniform limits in Alp band-dominated. To state a collection of important properties of Alp and its elements we let BUC denote the algebra of all bounded and uniformly continuous functions on the real line. For ϕ ∈ BUC, t > 0 and r ∈ R we define with ϕt (x) := ϕ(tx) and ϕt,r (x) := ϕt (x − r) certain inflated and shifted versions of ϕ. Furthermore, let ϕˆ stand for the sequence (ϕ(n))n∈Z which obviously belongs to l∞ , and therefore induces a multiplication operator ϕI. ˆ The following results are well known, and can be found in [63], Theorem 2.1.6 and Propositions 2.1.7ff, and in [44], Theorem 1.42. Theorem 1.55. . 1. The set Alp forms a closed and inverse closed subalgebra of L(lp , P) and K := K(lp , P), the set of all P-compact operators, is a closed ideal in Alp . Furthermore, Alp /K is inverse closed in the quotient algebra L(lp , P)/K. 2. For an operator A ∈ L(lp ) the following are equivalent: • A is band-dominated. • For every > 0, there exists an M > 0 such that whenever F, G are subsets of Z with dist(F, G) := inf{kx − yk : x ∈ F, y ∈ G} > M, then kχF AχG IkL(lp ) < . • For every ϕ ∈ BUC, lim sup k[A, ϕˆt,r I]k = 0. t→0 r∈R

• For every ϕ ∈ BUC,

lim k[A, ϕˆt I]k = 0.

t→0

We want to add another characterization of band-dominated operators. Although there is no known reference in the literature where these observations can be found in the present form, their proofs are based on well known ideas. We consider the continuous piecewise linear splines ϕ and ψ of norm one defined by ( ( 1 : |x| ≤ 43 1 : |x| ≤ 13 , ψ(x) = , ϕ(x) = 0 : |x| ≥ 32 0 : |x| ≥ 45 and their inflated and shifted copies ϕα,R and ψ α,R , where f α,R (x) := f (x/R − α). Proposition 1.56. The mapping SR : L(lp , P) → L(lp , P), given by X SR (A) := ϕˆα,R Aψˆα,R I,

(1.18)

α∈Z

is well defined and linear. The series (1.18) converges P-strongly, and SR (A) is a band operator with kSR (A)k ≤ 2kAk for every A ∈ L(lp , P) and R ∈ N.

Proof. For the spaces lp with 1 ≤ p < ∞ and for (l∞ )0 see [63], Proposition 2.2.2 or [44], Lemma 3.14. In order to prove the P-strong convergence on l∞ we take SR (A) ∈ L((l∞ )0 , P) and note that the sequence (SR (A)Pn )n∈N has a P-strong limit in L(l∞ , P) since l∞ is complete w.r.t. P-strong convergence, and by the arguments of the 9th step in the proof of Theorem 1.27 on page 20. Since χF SR (A)χG I = 0 for arbitrary subsets F, G of Z with dist(F, G) ≥ 2R, we find that SR (A) is banded (see [44], Proposition 1.36).

36


Notice that in case of a band operator A, the equality SR (A) = A holds for sufficiently large R due to the fact that ϕˆα,R Aψˆα,R I = ϕˆα,R A for all α and large R. Let A be a band-dominated operator. Then, by definition, there is a sequence (Am ) of band operators which tends to A in the norm. One might assume that such Am can be constructed by taking the restrictions of the matrix representation of A to a finite number of diagonals. In general, this is not true, as [44], Remark 1.40 shows. Nevertheless, the sequence (SR (A))R∈N provides a canonical approximation of A by band operators, in analogy to the Fejér-Cesàro means for the approximation of continuous functions by trigonometric polynomials. Indeed, for every fixed m and sufficiently large R, kSR (A) − Ak ≤ kSR (A − Am )k + kSR (Am ) − Am k + kAm − Ak ≤ 3kAm − Ak, where the latter becomes as small as desired if m is chosen sufficiently large. Conversely, if for an operator A ∈ L(lp ) the band operators SR (A) exist and converge to A in the norm, then A is band-dominated. Thus we have Corollary 1.57. An operator A ∈ L(lp ) is band-dominated, if and only if the operators SR (A) exist for every R ∈ N and kSR (A) − Ak → 0 as R → ∞. Concerning the stability of a finite section sequence (An ) = (Pn APn + Qn ) of A ∈ Alp we already know that the invertibility of A is a necessary condition.19 One may say that the single operator A, which is just the P-strong limit of (An ), captures one facet of the asymptotic behavior of this sequence. This seems to be a quite facile observation at a first glance, but nevertheless this is basically what we are going to do for the complete characterization of stability: Take snapshots of (An ) by passing to certain limit operators and draw on their invertibility. Of course, this simplest snapshot A describes the behavior at the center of the truncation intervals [−n, n]. The much more interesting points where the projections Pn have their major impact are at the boundary {−n, n}, and it seems to be natural to observe the behavior there as well. To do this one might replace (An ) by the shifted copies (A1n ) := (Vn An V−n ) and (A2n ) := (V−n An Vn ) which “fix one endpoint of the truncation intervals at the origin”. Figure 1.2 illustrates the movement of the focus. Clearly, the stability is invariant under these translations, and it is also obvious that every subsequence of a stable sequence is stable again. So, if (A1n ), (A2n ), or at least certain subsequences of them converge P-strongly then the invertibility of these limits is also necessary for the stability of the sequence (An ). Now, this immediately imposes the question if, with these additional snapshots, we have enough for the characterization of the stability of (An ), in the sense that their simultaneous invertibility gives a sufficient condition, or if we need any more. The answer will be “Yes, under a natural condition, it suffices to observe the sequence at its center and at the boundaries of the truncation interval.” Also for the asymptotics of the pseudospectra of An we already know from the basic example A = V (see Section 1.4.4) that the operator A alone is not enough to capture the limiting set, and we may guess that the additional snapshots could bring more light into the darkness. Incited by these problems we are going to examine the notion of rich operators in the next step, and to elaborate this sequence algebra approach in Part 2, of course in a more general and more abstract setting. There, we also discuss a Fredholm theory for sequences which extends the connection between stability and invertibility to certain connections between “almost stability” of a sequence and the Fredholm properties of its snapshots. Part 3 will treat also the question on spectral approximation. 19 Cf.

Proposition 1.29

1.5. EXAMPLE: OPERATORS ON LP -SPACES

37

n

An

Pm

Vn An V−n

Pm

Figure 1.2: P-strong limits of the sequence (An ) and its shifted copy (Vn An V−n ).

1.5.2

Rich operators

We already discovered that it might be fruitful to examine the asymptotics of shifted copies (V−gn AVgn ) of A ∈ L(lp , P), where gn is a sequence of integers whose absolute values tend to infinity. If such a sequence converges P-strongly (let’s say, to Ag ) then we call Ag the limit operator of A w.r.t. g. The set σop (A) of all limit operators of A is referred to as its operator spectrum. Further, we say that A is a rich operator if every sequence g of integers whose absolute values tend to ∞ has a subsequence h such that Ah exists. The theory of limit operators has been intensively studied during the last years and has a wide range of applications. We only touch this concept here but, nevertheless, we state some of its highlights. For a nice introduction and a comprehensive discussion consult the work of Lindner (e.g. [44], [15]) or the fundamental book of Rabinovich, Roch and Silbermann [63]. Theorem 1.58. Let A ∈ Alp be rich. Then A is P-Fredholm if and only if all limit operators of A are invertible and their inverses are uniformly bounded. This theorem has a long history and we particularly mention the pioneering paper [42] of Lange and Rabinovich. The proof of the if part is based on a construction of a P-regularizer, which goes back to Simonenko [86] and can be found in [62] and [44], for example.20 The only if part was discussed in [62] and [63], Theorem 2.2.1 (for 1 < p < ∞), in [44] (all p and with an additional assumption on the existence of a predual setting in the case p = ∞), and in [15], Theorem 6.28 (all p). We mention that the present approach provides this implication even for every rich operator A ∈ L(lp , P): Proof. Let g be such that Ag exists and let B be a P-regularizer for A. It is quite obvious from the definition that the operator spectrum of every P-compact operator K is trivial: σop (K) = {0}. Thus, besides V−gn AVgn → Ag , we also have V−gn (AB − I)Vgn → 0 and V−gn (BA − I)Vgn → 0 P-strongly. Then, for every T ∈ K(lp , P) kT k = kV−g(n) IVg(n) T k ≤ kV−g(n) BVg(n) kkV−g(n) AVg(n) T k + kV−g(n) (I − BA)Vg(n) T k, and consequently (for a certain constant D > 0 independent of g, and n → ∞) kT k ≤ DkAg T k for all T ∈ K(lp , P). The dual estimate kT k ≤ DkT Ag k for all T ∈ K(lp , P) follows analogously. Due to the Pdichotomy, this implies the invertibility of Ag and Theorem 1.14 yields that A−1 belongs to g L(lp , P). 20 Actually,

also our mappings SR are inspired by this construction.

38


For every y ∈ (lp )0 there is a projection Ry ∈ K(lp , P) with norm not greater than 2 onto span{y} (by Proposition 1.33), and therefore the first of the above estimates yields for the p −1 −1 −1 operator A−1 g Ry ∈ K(l , P) that kAg Ry k ≤ DkAg Ag Ry k ≤ 2D. Hence, kAg |(lp )0 k ≤ 2D and with (1.6) we get kA−1 k ≤ 2D with D independent of g. This yields the uniform boundedness g of the inverses. We also note that every limit operator of a band-dominated operator is again band-dominated (cf. [44], Proposition 3.6). Proposition 1.59. The set L$ (lp , P) of all rich operators is a closed subalgebra of L(lp , P) and contains K(lp , P) as a closed ideal. Every P-regularizer of a rich P-Fredholm operator is rich. In particular, L$ (lp , P) is inverse closed in L(lp , P) and L(lp ).21 Proof. It is obvious from the definition that L$ (lp , P) is a subalgebra of L(lp , P). Now, let (Ak ) ⊂ L$ (lp , P) be a sequence which tends uniformly to an operator A ∈ L(lp , P), and let h be a sequence of integers whose absolute values tend to ∞. Choose a subsequence g 1 ⊂ h such that A1g1 exists, a subsequence g 2 ⊂ g 1 such that A2g2 exists, a subsequence g 3 ⊂ g 2 for A3 , and so on. Then define a new sequence g = (gn ) by gn := gnn . Evidently, g is a subsequence of h and all m limit operators Am g , m ∈ N exist. One now easily checks that (Ag )m converges uniformly, and $ p its limit Ag is the limit operator of A w.r.t. g. Thus, L (l , P) is closed. Let A ∈ L$ (lp , P) be P-Fredholm, B be a P-regularizer, and g such that Ag exists. From the previous proof we already know that Ag is invertible, and we claim that Bg exists and equals A−1 g . Then it is clear that B is rich. Indeed, since V−g(n) BVg(n) − A−1 g

−1 = V−g(n) BVg(n) (I − V−g(n) AVg(n) A−1 g ) + V−g(n) (BA − I)Vg(n) Ag

−1 = V−g(n) BVg(n) (Ah − V−g(n) AVg(n) )A−1 g + V−g(n) (BA − I)Vg(n) Ag

p we get k(V−g(n) BVg(n) − A−1 g )T k → 0 for every T ∈ K(l , P). Analogously, we can show that )k → 0 and obtain the P-strong convergence of V−g(n) BVg(n) to A−1 kT (V−g(n) BVg(n) − A−1 g . g

Corollary 1.60. If dim X < ∞ then every band-dominated operator on lp = lp (Z, X) is rich. Proof. It suffices to consider operators aI of multiplication by a = (an ) ∈ l∞ (Z, L(X)), since the shift operators are obviously rich, and L$ (lp , P) is a closed algebra. With k := dim X and a fixed basis in X we see that the matrix representations of the (uniformly bounded) operators an are k × k-matrices and from the Bolzano-Weierstrass theorem we deduce the existence of a convergent subsequence. For general X we have Theorem 1.61. (see [63], Theorem 2.1.16) The operator aI of multiplication by a = (an ) ∈ lp (Z, L(X)) is rich if and only if the set of its values {an : n ∈ Z} is relatively compact (with respect to the operator norm on L(X)).

21 The notion of rich operators can be introduced in much more general settings L(X, P), where a certain family of “shift-like” operators is available (see [63], Section 1.2 for details). If X has the P-dichotomy then this proposition is valid again and generalizes the results of [63], Section 1.2.1.

Part 2

Fredholm sequences and approximation numbers We now turn our attention to algebras of operator sequences A = {An } which have a certain asymptotic structure in common. More precisely, for every sequence there shall be a family of operators W t (A) which appear as P-strong limits (of the sequence A or slightly modified copies). The goal of Sections 2.1 and 2.2 will be a Fredholm theory for such sequences, which provides a couple of relations between the properties of A (such as stability or Fredholmness), properties of its entries An (e.g. the asymptotic behavior of the approximation numbers or indices), and the Fredholm properties of its so-called snapshots W t (A). Section 2.3 is devoted to a notion of Fredholm sequences which does not require any asymptotic structure and which can be regarded as a generalization of the first concept. Finally, we deal with sequences which are somehow in between of these two approaches, so-called rich sequences. They do not have an asymptotic structure as considered in the first section, but they have at least “sufficiently many” subsequences of that structured type. We apply the established tools to band-dominated operators in Section 2.5. Such concepts of Fredholmness for operator sequences have their origin in the consideration of the finite section method for Toeplitz operators [83], are well sophisticated in the framework of Standard Algebras [30], that are classes of C ∗ -algebras, have been studied in the case of matrix sequences A with ∗-strong limits W t (A) [75] and there are also variants for special classes of operator sequences in a Banach space context, again with the classical notions of compactness and convergence [50]. The fusion with the concept of approximate projections was first done by the author in [80] for matrix and in [81] for operator sequences. Clearly, these latest releases are heavily influenced by its forerunners and include or adapt several former ideas. The reader will particularly rediscover the golden thread of [75] in the presentation of the Sections 2.1 and 2.2. The content of the Sections 2.1 to 2.3 and 2.5 is subject of the mentioned publications [80], [81], but here in a complete and stand-alone form, whereas the notion of rich sequences appears here for the first time and generalizes the notion of rich operators 1 and concrete tools for banddominated operators, in a sense.

1 Cf.

Section 1.5.2.

40

2.1

PART 2. FREDHOLM SEQUENCES AND APPROXIMATION NUMBERS

Sequence algebras

Let (En ) be a sequence of Banach spaces and let Ln stand for the identity operator on En , respectively. We denote by F the set of all bounded sequences {An }2 of bounded linear operators An ∈ L(En ). One easily verifies that, provided with the operations α{An } + β{Bn } := {αAn + βBn },

{An }{Bn } := {An Bn },

and the supremum norm k{An }kF := supn kAn kL(En ) < ∞, F becomes a Banach algebra with identity I := {Ln }. The set G := {{Gn } ∈ F : kGn kL(En ) → 0 as n → ∞} forms a closed ideal in F. Further, let T be a (possibly infinite) index set and suppose that, for every t ∈ T , there is a Banach space Et with a uniform approximate identity P t such that Et has the P t -dichotomy, and a bounded sequence (Ltn ) of projections Ltn ∈ L(Et , P t ) tending P t -strongly to the identity I t on Et . Set ct := sup{kLtn kL(Et ) : n ∈ N} < ∞ for every t ∈ T. Suppose that, for every t ∈ T , there is a sequence (Ent ) of algebra isomorphisms Ent : L(im Ltn ) → L(En ), such that (for brevity, we write En−t instead of (Ent )−1 ) M t := sup{kEnt k, kEn−t k : n ∈ N} < ∞.

(I)

We denote by F T the collection of all sequences A = {An } ∈ F, for which there exist operators W t (A) ∈ L(Et , P t ) for all t ∈ T such that −t t t t A(t) n := En (An )Ln → W (A) P -strongly.

These limits are uniquely determined and with the help of Theorem 1.19 it is easy to show that F T is a closed subalgebra of F which contains the identity and the ideal G. Both, the mappings Ent and W t : F T → L(Et , P t ), A 7→ W t (A) are unital homomorphisms for every t ∈ T . Roughly speaking, the mappings Ent allow us to transform a given sequence A ∈ F T and to generate snapshots W t (A) from different angles which outline several aspects of the asymptotic behavior of A. In what follows, we will examine the connections between the properties of A and the properties of its “snapshots at infinity”. Remark 2.1. The results and proofs of the subsequent sections remain true, if for some or all t ∈ T the sequence (Ltn ) of projections converges ∗-strongly to the identity, and if we replace P t -Fredholmness by Fredholmness, P t -compactness by compactness, and P t -strong convergence by ∗-strong convergence. An approximate projection P t is not needed in this case. Thus, the results for so-called Standard Algebras (see [30]) and for Banach algebras of matrix sequences in [75] are completely covered by the present considerations. Furthermore, the generalization of the approach of [75] to sequences of operators acting on infinite dimensional Banach spaces in [50] is a special case of the theory in the present text as well. 2 We continue to use (·) for sequences of elements in one common space, whereas the sequences in F which consist of elements coming from different spaces En in general are denoted by {·}.

2.2. J T -FREDHOLM SEQUENCES

41

An

.

L(En ) Ent1 E t1 n+1 t1

L(E )

(t )

An 1

t1

...

L(En+1 ) ...

...

Ent2

P

. An+1

...

n

t1 t2

L(E )

W (A)

...

(t )

An 2

t2

...

P

n

t2

W (A)

Figure 2.1: Snapshots of a sequence A = {An } ∈ F T .

2.2

J T -Fredholm sequences

Here we introduce the notion of Fredholm sequences. A sequence will be called Fredholm, if it is “invertible modulo an ideal of compact sequences”, where in our first approach these “compact sequences” will be generated by lifting P t -compact operators. For this, it turns out to be reasonable to suppose that the perspectives from which a sequence can be looked at, and which are given by the homomorphisms W t , are different from each other. Therefore, and in all what follows, we suppose that the separation condition W

τ

{Ent (Ltn K t )}

( Kt = 0

if t = τ if t 6= τ

(II)

holds for all τ, t ∈ T and every K t ∈ Kt := K(Et , P t ). Metaphorically speaking, this means that the directions from which one can look at a sequence are separated in the sense that the P t -compact operators K t and their liftings {Ent (Ltn K t )} which arise from one point of view t ∈ T are invisible from every other direction. Definition 2.2. We put

3

J t := {{Ent (Ltn K t )} + {Gn } : K t ∈ Kt , kGn k → 0} (∀ t ∈ T ), (m ) X T ti ti ti J := closF T {Jn } : m ∈ N, ti ∈ T, {Jn } ∈ J . i=1

One may refer to the elements of these sets as compact sequences. Actually, we will see that this is entirely justified since the algebraic structure is similar to the notions of compact or P-compact operators. We start with the following observation. Proposition 2.3. All J t , t ∈ T as well as J T are closed ideals in F T . 3 The consideration of such ideals goes back to the approach of Silbermann for the finite section method applied to Toeplitz operators in [83]. See also Theorem 2.17 and the annotations there.

42


Proof. For t ∈ T , K ∈ Kt and A = {An } ∈ F T we have t t t t t (t) t An Ent (Ltn K) = Ent (A(t) n K) = En (Ln W (A)K) + En (Ln (An − W (A))K),

t t t t t (t) t Ent (Ltn K)An = Ent (Ltn KA(t) n ) = En (Ln KW (A)) + En (Ln K(An − W (A))),

where W t (A) ∈ L(Et , P t ) and hence W t (A)K, KW t (A) ∈ Kt . Due to the Condition (I) and (t) since (An − W t (A)) tends P t -strongly to 0, the last summands tend to zero in the norm as n → ∞ in both cases. Consequently, all J t as well as J T are ideals in F T , where J T is closed by its definition. Let ({Jnk })k∈N = ({Ent (Ltn Kkt )} + {Gkn })k∈N ⊂ J t be a Cauchy sequence. From Theorem 1.19 we obtain (with C t := ct M t BP t ) kKkt − Klt k = kW t {Ent (Ltn (Kkt − Klt ))} + {Gkn } − {Gln } k ≤ C t k({Ent (Ltn Kkt )} + {Gkn }) − ({Ent (Ltn Klt )} + {Gln })k = C t k{Jnk } − {Jnl }k,

therefore the sequence (Kkt )k∈N is a Cauchy sequence in Kt , and since Kt is closed it possesses a limit K t ∈ Kt . Analogously, the estimate k{Gkn } − {Gln }k ≤ k({Ent (Ltn Kkt )} + {Gkn }) − ({Ent (Ltn Klt )} + {Gln }) + k{Ent (Ltn (Klt − Kkt ))}k ≤ k{Jnk } − {Jnl }k + M t ct kKlt − Kkt k

shows that {Gkn } converges to a certain {Gn } ∈ G. Now it’s easy to see that the sequence {Ent (Ltn K t )} + {Gn } ∈ J t is the limit of ({Jnk })k and the closedness of J t is proved. Definition 2.4. A sequence A ∈ F T is said to be J T -Fredholm, or regularizable with respect to J T , if the coset A + J T is invertible in the quotient algebra F T /J T . Notice that this property depends on the underlying algebra F T and the ideal J T . It is obvious that the set of J T -Fredholm sequences is open in F T , the sum of a J T -Fredholm sequence and a sequence from the ideal J T is J T -Fredholm and that the product of two J T -Fredholm sequences is J T -Fredholm again. We will see that there are much deeper similarities with the usual Fredholm property of operators. In particular, there are analogues to the kernel and cokernel dimension as well as the index. Let us start with a basic result on the regularization of a J T -Fredholm sequence. Proposition 2.5. Let A ∈ F T be J T -Fredholm. Then there exist finite subsets {t1 , ..., tm } and {τ1 , ..., τl } of T and a δ > 0 such that the following holds: ˜ ∈ F T with kA − Ak ˜ < δ there are sequences B, C ∈ F T and G, H ∈ G as well as For each A ti ti τi operators K ∈ K and K ∈ Kτi such that ˜ =I+ BA

m X i=1

˜ =I+ AC

l X i=1

{Enti (Ltni K ti )} + G,

(2.1)

{Enτi (Lτni K τi )} + H.

(2.2)

ˆ ∈ FT , Proof. By the definitions of J T -Fredholmness and of the ideal J T , there exist a sequence A t t τ τ ˆ i ∈ J i as well as finite subsets {t1 , ..., tm } and {τ1 , ..., τl } of T and sequences ˆJ i ∈ J i , K


43

ˆ ˆ ∈ J T with kˆ ˆ < 1/4 such that J, K Jk, kKk ˆ =I+ AA

m X

ˆ =I+ ˆ Jti + ˆJ and AA

i=1

l X

ˆ ˆ τi + K. K

i=1

˜ ∈ F T with kA ˜ − Ak < δ := 1/(4kAk) ˆ we have For A Â ˜ =I+ A

m X i=1

ˆ A ˜ − A) ˆJti + ˆJ + A(

ˆ A ˜ − A)k < 1/2. Hence, the sequence I + ˆJ + A( ˆ A ˜ − A) is invertible in the Banach where kˆJ + A( T algebra F and we can define ˆ A ˜ − A)]−1 A ˆ ∈ F T and B := [I + ˆ J + A( ˆ A ˜ − A)]−1 ˆJti ∈ J ti J + A( Jti := [I + ˆ for i = 1, ..., m. Due to the definition of the ideals J t , there are operators K ti ∈ Kti and a sequence G ∈ G such that ˜ =I+ BA

m X

m X {Enti (Ltni K ti )} + G.

Jti = I +

i=1

i=1

Analogously, one obtains sequences C ∈ F T and H ∈ G as well as operators K τi ∈ Kτi such that Eq. (2.2) holds. Applying the homomorphisms W t , t ∈ T to the Equations (2.1) and (2.2), and utilizing the separation condition, we see that the following theorem is in force. Theorem 2.6. If a sequence A ∈ F T is J T -Fredholm, then all corresponding operators W t (A) are P t -Fredholm on Et and the number of the non-invertible operators among them is finite.

2.2.1

Systems of operators

For fixed t ∈ T let P t ∈ Kt be a P t -compact operator. We set Pˆ t := I t − P t and obtain I t − (I t − Ltn )P t = Ltn P t + Pˆ t . Since P t ∈ Kt , there is an nt ∈ N, such that k(I t − Ltn )P t k, kP t (I t − Ltn )k < 1 for all n ≥ nt . Thus, for n ≥ nt , we obtain the invertibility of I t − (I t − Ltn )P t and, moreover, the inverses are (I t − (I t − Ltn )P t )−1 =

∞ X

((I t − Ltn )P t )k ∈ L(Et , P t ).

k=0

This justifies the following definition for n ≥ nt : I t = Ltn P t |

∞ X

k=0

((I t − Ltn )P t )k + Pˆ t {z

=:Pnt

}

|

∞ X

((I t − Ltn )P t )k .

k=0

{z

=:Pˆnt

}

44


For the operators Pnt we have im Pnt ⊂ im Ltn and kP t − Pnt k tends to 0 as n → ∞ since kP t − Pnt k = kP t − I t + I t − Pnt k = kPˆnt − Pˆ t k

∞

X k(I t − Ltn )P t k

ˆt t t t k ((I − Ln )P ) ≤ kPˆ t k = P → 0.

1 − k(I t − Ltn )P t k k=1

Applying the homomorphisms Ent , we can lift them to a sequence {Rnt } ∈ F T by the rule ( Ent (Pnt ) : n ≥ nt t . Rn := 0 : n < nt Indeed, it is easy to check that {Rnt } is bounded, and it belongs to J t because of kRnt − Ent (Ltn P t )k = kEnt (Pnt − Ltn P t )k ≤ M t (kPnt − P t k + k(I t − Ltn )P t k) → 0. One can further show that, for n large, dim im Rnt = dim im Pnt ≤ dim im P t and, for every k < dim im P t , there is an Nkt with dim im Rnt > k whenever n ≥ Nkt . Definition 2.7. For t ∈ T let P t ∈ Kt . The system (P t , Pˆ t , Pnt , Pˆnt , Rnt ) of operators (with n ≥ nt ), which we can construct as above, is called a P t -corresponding system of operators. The following proposition shows that for sequences A ∈ F T , applying such P t -corresponding systems, certain properties of boundedness of the operators W t (A) can be devolved on the sequence A. Proposition 2.8. Let A = {An } ∈ F T , P t ∈ Kt be a P t -compact operator with the P t corresponding system (P t , Pˆ t , Pnt , Pˆnt , Rnt ). Then lim sup kAn Rnt k ≤ M t kW t (A)P t k,

and

n→∞

lim sup kRnt An k ≤ M t kP t W t (A)k. n→∞

Proof. From the above considerations we get t t (t) t kAn Rnt k = kAn Ent (Pnt )k = kEnt (A(t) n Pn )k ≤ M kAn Pn k

t t t t t t t t ≤ M t (kA(t) n P k + M c kAkkPn − P k) → M kW (A)P k,

(t)

since An → W t (A) P t -strongly and P t is P t -compact. The dual assertion can be checked analogously. Moreover, if P t is a projection, then the relations Pˆ t Pˆnt = Pˆ t Pˆ t

∞ X

((I t − Ltn )P t )k = Pˆ t

k=0

Pˆnt Pˆ t = Pˆ t

∞ X

∞ X

((I t − Ltn )P t )k = Pˆnt ,

k=0

((I t − Ltn )P t )k Pˆ t = Pˆ t I t Pˆ t + 0 = Pˆ t , and

k=0

Pˆnt Pˆnt = Pˆnt (Pˆ t Pˆnt ) = (Pˆnt Pˆ t )Pˆnt = Pˆ t Pˆnt = Pˆnt

imply that also the operators Pnt ∈ Kt and Rnt ∈ L(En ) are projections and one can easily check that dim im Rnt = dim im Pnt = dim im P t for sufficiently large n.


2.2.2

45

The approximation numbers of An and the snapshots W t {An }

Here is a further connection between the operators An of a sequence {An } ∈ F T and its snapshots W t {An }.

Theorem 2.9. Let A = {An } ∈ F T , m ∈ N and all W ti (A) are Pmt1 , . . . , tm ∈ tiT be such that l r dim ker W (A), and s (A P ti -Fredholm. Then s (A ) → 0 for all k ≤ n ) → 0 for all n k k i=1 Pm k ≤ i=1 dim coker W ti (A) as n → ∞. If, for one t ∈ T , the snapshot W t (A) is properly P t -deficient from the right (from the left) then srk (An ) → 0 (or slk (An ) → 0, respectively) for every k ∈ N as n → ∞. Proof. Proposition 1.32 reveals that, if all of these W ti (A) are normally solvable then, for each i = 1, ..., m and every respective non-negative integer ki ≤ dim ker W ti (A), we can fix a P ti compact projection P i onto a ki -dimensional subspace of ker W ti (A) and choose the respective system (P t , Pˆ t , Pnt , Pˆnt , Rnt ) of projections. Moreover, for each i and every n ≥ maxj ntj we n i choose a normed basis {xni,l }kl=1 of im Rni , such that for arbitrary scalars αi,j the following holds:

X

ki n n n

(2.3) αi,j xi,j |αi,p | ≤

for all p = 1, ..., ki .

j=1

It is a simple consequence of Auerbach’s Lemma (see Proposition 1.4) that such a basis always exists. Let i 6= j. Due to the separation condition (II), since kPnj − P j k → 0 and P j ∈ Ktj , we have kRni Rnj k = kEnti (Pni )Entj (Pnj )kL(En ) ≤ M tj kEn−tj (Enti (Pni ))Pnj k → 0.

(2.4)

We now prove that there is a number Tm N ∈ N, such that, for all j = 1, ..., m, k = 1, ..., kj , n ≥ N , all scalars αi,l , and all y ∈ Yn := i=1 ker Rni

ki m X

X

n αi,l xi,l + y , where γ = 2 max M ti kP i k. (2.5) |αj,k | ≤ γ i=1,...,m

i=1 l=1

In other words, all of these xni,j are linearly independent in a “uniform” manner. By (2.4), for every ∈ (0, 1), there exists a number N ∈ N such that m X

j=1,j6=i

kj kRni Rnj k ≤ and kRni k ≤ M ti kPni k ≤ (1 + )M ti kP i k

(2.6)

for all i = 1, ..., m and n ≥ N . Now let αi,l be arbitrary but fixed scalars and choose i0 , l0 such that |αi0 ,l0 | = maxi,l |αi,l |. Thanks to (2.3) and (2.6) we find (since xni,j = Rni xni,j )

m k

m

ki0 ki i

X X

X X X

i0 n i0 n i0 n

kRn k αi,l xi,l + y ≥ αi0 ,l Rn xi0 ,l − αi,l Rn xi,l

i=1,i6=i0 l=1

i=1 l=1 l=1

ki0

ki m X X

X

n

≥ αi0 ,l xi0 ,l − |αi,l |kRni0 Rni xni,l k

l=1

i=1,i6=i0 l=1 ≥ |αi0 ,l0 | − |αi0 ,l0 |

≥ (1 − )|αi0 ,l0 |.

m X

i=1,i6=i0

ki kRni0 Rni k

46


Thus, for all n ≥ N , for all j = 1, ..., m, and all k = 1, ..., kj ,

m k

i

kRni0 k

X X

αi,l xni,l + y |αj,k | ≤ |αi0 ,l0 | ≤

1 − i=1 l=1

m ki

1+

X X

ti i n ≤ max M kP k · αi,l xi,l + y .

1 − i=1,...,m i=1 l=1

Choosing = 1/3 gives assertion (2.5). Obviously, En decomposes into the direct sum En = span{xn1,1 } ⊕ . . . ⊕ span{xnm,km } ⊕ Yn n ∈ E∗n by the rule for n ≥ N , and we can introduce functionals fi,j n fi,j

kk m X X

n αk,l xnk,l

+y

k=1 l=1

!

n := αi,j 1 ≤ i ≤ m, 1 ≤ j ≤ ki .

n n Then we always have kfi,j k ≤ γ and fi,j (Yn ) = {0}. Now, we denote by Rn ∈ L(En ) the linear operators ki m X X n fi,j (x)xni,j . Rn x := i=1 j=1

Pm The operators Rn are projections of rank dim im Rn = k := i=1 ki and they are uniformly bounded with respect to n. Moreover, for every x ∈ En we have

ki ki m X m X

X

X n n n

kAn Rn xk = An fi,j (x)xi,j ≤ |fi,j (x)|kAn Rni xni,j k

i=1 j=1 i=1 j=1 ≤

ki m X X i=1 j=1

γkxkkAn Rni kkxni,j k = γkxk

m X i=1

ki kAn Rni k.

Since kAn Rni k → 0 for each i (see Proposition 2.8) it follows that srk (An ) = inf{kAn + F k : F ∈ L(En ), dim ker F ≥ k}

≤ kAn − An (Ln − Rn )k = kAn Rn k ≤ γk max kAn Rni k → 0. i

Due to Proposition 1.32 we can also choose P ti -compact projections P˜ ti with P˜ ti W ti (A) = 0 n ˜ n, ˜ ni and Y and proceed in the same way to find the analogues k˜i ≤ dim coker W ti (A), x ñi,j , f˜i,j ,R Pm ˜ ˜ ˜ and to construct a bounded sequence (Rn ) of projections Rn , being of rank k = i=1 dim k˜i . Then

X

X k˜i k˜i m X

m X

n n n ˜i n ˜ ˜ ˜ ni )An x)|k˜

kRn An xk = fi,j (An x)˜ xi,j ≤ |f˜i,j (Rn An x) + f˜i,j ((I − R xni,j k

i=1 j=1

i=1 j=1 ≤

k˜i m X X i=1 j=1

n n ˜ ni An k + kf˜i,j ˜ ni )kkAn || kxk, kf˜i,j kkR (I − R


47

n ˜ n we have ˜ ni )k → 0 as n → ∞ due to the following observations: for each y ∈ Y (I − R where kf˜i,j i n i n n ˜ ˜ ˜ ˜ ˜ ˜ fi,j ((I − Rn )y) = fi,j (y) = 0, that is fi,j (I − Rn )(I − Rn ) = 0, and further n ˜ i )R ñk ≤ kf˜i,j (I − R n

k˜s m X X s=1 l=1

n n n ˜ i )˜ |f˜s,l |kf˜i,j (I − R n xs,l k → 0

n n ˜i n n ˜ ni )˜ ˜ ni )˜ ˜ ni R ˜ ns kk˜ since (I − R xni,l = 0 and kf˜i,j (I − R xns,l k = kf˜i,j Rn x ˜s,l k ≤ kf˜i,j kkR xns,l k → 0 for s 6= i as n → ∞. Thus, we again obtain

˜ n An k → 0 as n → ∞. slk (An ) ≤ kR Now, suppose that one operator W t (A) is properly P t -deficient from the right (or left). Then for each k ∈ N and each > 0 there is a projection Q ∈ Kt , rank Q = k such that kW t (A)Qk < (or kQW t (A)k < , respectively). Choosing a system of projections w.r.t. Q, we deduce again from Proposition 2.8 that lim supn srk (An ) = 0 (or lim supn slk (An ) = 0) since can be chosen arbitrarily small. Finally notice that W ti (A) being P ti -Fredholm but not normally solvable implies P ti -deficiency from both sides by Proposition 1.32.

2.2.3

Regular J T -Fredholm sequences

Definition 2.10. We introduce the nullity α(A) and deficiency β(A) of a sequence A ∈ F T by α(A) :=

X

dim ker W t (A) and β(A) :=

t∈T

X

dim coker W t (A).

t∈T

A J T -Fredholm sequence A ∈ F T is said to be regular, if all operators W t (A) are Fredholm (in the usual sense). Of course, due to Theorem 2.6, A is regular if and only if α(A) and β(A) are finite, hence we are in a position to introduce the index of a regular sequence A by 4 ind A := α(A) − β(A). Applying Proposition 2.5 and the well known properties of Fredholm operators as stated in Theorem 1.2 it is not hard to prove the following result. Proposition 2.11. Let A ∈ F T be a regular J T -Fredholm sequence and B ∈ F T . • If kBk is sufficiently small then α(A + B) ≤ α(A), β(A + B) ≤ β(A) and ind(A + B) = ind(A). • If B ∈ J T has only compact snapshots then A + B is regular and ind(A + B) = ind A. • If B ∈ G then α(A + B) = α(A) and β(A + B) = β(A). • If B ∈ F T is also a regular J T -Fredholm sequence then ind(AB) = ind A + ind B. Observe that, if all approximate identities P t consist of finite rank operators only, then K(Et , P t ) is a subset of K(Et ), and hence every J T -Fredholm sequence is regular. 4 This

already appeared in [30] and [50].

48

2.2.4


The splitting property and the index formula

Proposition 2.12. Let A = {An } ∈ F T be a J T -Fredholm sequence and suppose that all of its snapshots W t (A) are properly P t -Fredholm operators, respectively. If α(A) is finite then lim inf n srα(A)+1 (An ) > 0, and if β(A) is finite then lim inf n slβ(A)+1 (An ) > 0. Proof. Assume that α(A) is finite. Due to Equation (2.1) BA = I +

m X {Enti (Ltni K ti )} + G i=1

t

from Proposition 2.5, all operators W (A) with t ∈ T \{t1 , ..., tm } have kernels of dimension zero. Moreover, for every i = 1, ..., m, Corollary 1.31 provides an operator B i ∈ L(Eti , P ti ) such that P i := I ti − B i W ti (A) ∈ Kti is a projection onto the kernel of W ti (A). Define Pˆ i := I ti − P i and furthermore, for every n ∈ N, (with {Bn } = B) Dn := Bn −

m X

Enti (Ltni K ti B i ).

i=1

(t )

It is obvious that {Dn } ∈ F T and due to the P ti -strong convergence of An i we see that

ti i) kEnti (Ltni K ti B i )An − Enti (Ltni K ti Pˆ i )k = kEnti (Ltni K ti B i (A(t n − W (A)))k → 0 Pm Pm for all i, i.e. { i=1 Enti (Ltni K ti B i )An − i=1 Enti (Ltni K ti Pˆ i )} ∈ G. Thus

Dn An = Bn An − = Ln +

m X

m X i=1

= Ln +

m X

Enti (Ltni K ti B i )An

i=1

Enti (Ltni K ti ) −

m X

Enti (Ltni K ti Pˆ i ) + Hn

i=1

Enti (Ltni K ti P i ) + Hn

i=1

i

with {Hn } ∈ G. Since dim im P = dim ker W ti (A) for all i, we find ! m X dim im Enti (Ltni K ti P i ) ≤ α(A). i=1

For sufficiently large n we have kHn k < 1/2 and, applying Corollary 1.37, it follows 1 ≤ (k(Ln + Hn )−1 k)−1 = sr1 (Ln + Hn ) 2 = inf{kLn + Hn − F k : dim ker F ≥ 1}

( ) m

X

ti ti ti i ≤ inf Ln + Hn − F + En (Ln K P ) : dim ker F ≥ α(A) + 1

i=1

= inf{kDn An − F k : dim ker F ≥ α(A) + 1}

≤ inf{kDn An − Dn F k : dim ker F ≥ α(A) + 1}

≤ kDn k inf{kAn − F k : dim ker F ≥ α(A) + 1} ≤ k{Dn }ksrα(A)+1 (An ). If we start with Equation (2.2) we can proceed analogously to obtain slβ(A)+1 (An ) ≥ const > 0 for all sufficiently large n.


49

We have that all operators W t (A) of a regular J T -Fredholm sequence A are Fredholm and P t Fredholm, hence properly P t -Fredholm by Proposition 1.26. In view of Theorem 2.9 this yields the following result. Theorem 2.13. Let A = {An } ∈ F T be a regular J T -Fredholm sequence. Then the approximation numbers from the right have the α(A)-splitting property, that is lim sr (An ) n→∞ α(A)

=0

and

lim inf srα(A)+1 (An ) > 0. n→∞

Analogously, the approximation numbers from the left have the β(A)-splitting property. Theorem 2.14. Let A = {An } ∈ F T be a regular J T -Fredholm sequence. Then, for sufficiently large n, the operators An are Fredholm and their indices coincide with ind A, in other words lim ind An =

n→∞

X

ind W t (A).

t∈T

Proof. Theorem 2.13 shows that srα(A)+1 (An ) > 0 and slβ(A)+1 (An ) > 0 for sufficiently large n, hence An is Fredholm by Corollary 1.38. Moreover, for large n there is always an operator Fn with dim coker Fn ≥ β(A) such that kAn − Fn k ≤ slβ(A) (An ) + 1/n. We can choose a projection Sn of rank β(A) with norm less than β(A) + 2 such that kSn Fn k < 1/n (see Proposition 1.6) and we deduce from Theorem 2.13 that kSn An k kSn (An − Fn )k + kSn Fn k kSn An k 2 < ≤ ≤ slβ(A) (An ) + →n→∞ 0. β(A) + 2 kSn k kSn k n We set Sn := 0 for the remaining smaller n and conclude that {Sn An } ∈ G, that is the sequence {Añ } := {(Ln − Sn )An } ∈ F T is J T -Fredholm with α({Añ }) = α(A), β({Añ }) = β(A) and ind{Añ } = ind A (cf. Proposition 2.11). Analogously, we choose a sequence {Rn } ∈ F of projections Rn of rank α(A) and kRn k ≤ α(A) + 2 for large n, such that {Añ Rn } ∈ G. We now consider the sequence C = {Cn } := {(Ln − Sn )An (Ln − Rn )} and we find that it is a regular J T -Fredholm sequence with α(C) = α(A), β(C) = β(A) and ind C = ind A. More precisely, there is an N ∈ N and a constant C > 0 such that for all n ≥ N srα(A) (Cn ) = slβ(A) (Cn ) = 0 and srα(A)+1 (Cn ), slβ(A)+1 (Cn ) > C, which proves that the Cn are Fredholm of the index α(A) − β(A) = ind A. Since Ln − Rn and Ln − Sn are Fredholm of index zero, we find that An is Fredholm of index ind A. Remark 2.15. Note that there is a bounded sequence {Dn } such that for sufficiently large n the operator Dn is a generalized inverse for the Cn from the previous proof. Moreover, A − C belongs to G, hence we find that A + G is generalized invertible in F/G, whenever A is a regular J T -Fredholm sequence. Furthermore, this shows that the nullity, deficiency and the index of a structured operator sequence A being J T -Fredholm as introduced in Definition 2.10 are universal characteristics of this sequence in the following sense: If these numbers exist for the J T -Fredholm sequence A ˜ ˜ in one setting F T then they are the same in every setting F T in which A is J T -Fredholm. In Section 2.3 we will recover this observation from a more general point of view.

50

2.2.5


Stability of sequences A ∈ F T

Definition 2.16. A sequence A = {An } ∈ F is called stable, if there is an index n0 such that all operators An , n ≥ n0 , are invertible and sup{kA−1 n k : n ≥ n0 } < ∞. It is a well known result of Kozak [39] that a sequence A ∈ F is stable if and only if the coset A+G is invertible in F/G. Utilizing the higher structure of the given setting, namely the existence of P t -strong limits W t (A), we can prove a stronger result. This observation has a long history and is known as the Lifting Theorem in the literature, see for example the books [12], Section 7.8ff, [29], Section 1.6 or [30], Section 5.3. It has its origin in [83]. Theorem 2.17. A sequence A ∈ F T is stable if and only if A is J T -Fredholm and all W t (A), t ∈ T , are invertible. In particular, F T /G is inverse closed in F/G. Proof. Let A = {An } be J T -Fredholm and all W t (A) be invertible. Then A is regular and Theorem 2.13 yields an N ∈ N and a constant C > 0 such that sr1 (An ) ≥ C and sl1 (An ) ≥ C for all n > N . With Corollary 1.37 we deduce that A is stable. Conversely, let A = {An } ∈ F T be stable and define a sequence B = {Bn } ∈ F by Bn := A−1 n if An is invertible and Bn := Ln otherwise. Then BA − I, AB − I ∈ G. Notice that for every t ∈ T the sequence (Atn ) := (En−t (An )Ltn + (I t − Ltn )) ∈ F(Et , P t )

is also stable, tends to W t (A) and the operators Bnt := En−t (Bn )Ltn + (I t − Ltn ) are the inverses of Atn for sufficiently large n. Proposition 1.29 shows that W t (A) is invertible and (Bnt ) converges P t -strongly to (W t (A))−1 . Thus, B ∈ F T .

2.3

A general Fredholm property

The notion of J T -Fredholmness for structured operator sequences A ∈ F T as it was introduced in the preceding sections depends on the underlying setting determined by F T and J T . On the one hand, it is convenient to work in this context because there we can describe the Fredholm properties of a sequence A quite well in terms of Fredholm properties of its snapshots W t (A). On the other hand, the main disadvantage lies in the fact that a sequence which is Fredholm in one setting does not need to be Fredholm in another setting.5 Thus, in what follows, we introduce a “universal” Fredholm property for the larger framework of all bounded operator sequences in F. This approach has been extensively studied in the case of C ∗ -algebras of operator sequences on finite dimensional spaces in [30], Chapter 6.3. and was derived by Roch [70]. In particular, we will recover the mentioned universal characteristics α(A), β(A) and ind(A) in this larger framework again. To avoid degenerated cases we assume that lim supn dim En = ∞. Definition 2.18. A sequence {Kn } ∈ F is said to be of almost uniformly bounded rank if lim sup rank Kn < ∞. n→∞

Let I denote the closure of the set containing all sequences of almost uniformly bounded rank. The elements of I are referred to as compact sequences. One easily checks that I forms a proper closed ideal in F which contains G. 5 Clearly, A = I + {E t (Lt K t )} with K t ∈ K(Et , P t ) is J T -Fredholm, but if we drop t from the index set T n n ˜ and consider A as a sequence in F T with T˜ = T \ {t} then we loose this property.

2.3. A GENERAL FREDHOLM PROPERTY

51

Definition 2.19. Now we are in a position to introduce a class of Fredholm sequences in F by calling A = {An } ∈ F Fredholm if A + I is invertible in F/I. Evidently, we have the typical statements as known for the classical Fredholmness • Stable sequences are Fredholm and never compact. • Products of Fredholm sequences are Fredholm. • The sum of a Fredholm sequence and a compact sequence is Fredholm. • The set of all Fredholm sequences is open in F. • If {An } is Fredholm, then {A∗n } is of Fredholm type. For an equivalent characterization of Fredholm sequences we need the following definition. Definition 2.20. Let A = {An } ∈ F. If there is a finite number α ∈ Z+ with lim inf srα (An ) = 0 and lim inf srα+1 (An ) > 0, n→∞

n→∞

then this number is called the α-number of A and it is denoted by α(A). Analogously, we introduce β(A), the β-number of A, as the β ∈ Z+ with lim inf slβ (An ) = 0 and lim inf slβ+1 (An ) > 0. n→∞

n→∞

Besides the well known result of Kozak [39], by applying Corollary 1.37, we immediately get the following characterization of stability in the large algebra F. Theorem 2.21. For a sequence A ∈ F the following are equivalent. • A is stable. • A + G is invertible in F/G. • α(A) = β(A) = 0. Theorem 2.22. For a sequence A ∈ F the following are equivalent. • A is Fredholm. • There are sequences B1 , B2 ∈ F such that B1 A − I and AB2 − I are of almost uniformly bounded rank. • A has an α-number and a β-number. Proof. Let A = {An } be Fredholm. Then there are sequences D, Hi , Gi ∈ F (i = 1, 2) such that DA = I + H1 + G1 and AD = I + H2 + G2 , where Hi are of almost uniformly bounded rank and kGi k < 1/2. Since I + Gi are invertible, we can define B1 := (I + G1 )−1 D, K1 := (I + G1 )−1 H1 as well as B2 := D(I + G2 )−1 , K2 := H2 (I + G2 )−1 . This implies the second assertion. Now let {Kn } = {Bn }{An } − I and n0 ∈ N with k := supn≥n0 rank Kn < ∞. For each n ≥ n0 we can introduce a projection Rn with kernel of dimension k and norm kRn k ≤ k + 1 such that Rn Kn = 0, due to Proposition 1.5. Then Rn Bn An = Rn . Moreover, for each n ≥ n0 we observe that srk+1 (Rn ) ≥ 1, because otherwise there would exist an operator F with dim ker F ≥ k + 1

52


such that kRn − F k < 1. Hence Ln − Rn + F would be invertible, but since rank(Ln − Rn ) = k this yields a contradiction. Thus 1 ≤ srk+1 (Rn ) = inf{kRn − F k : dim ker F ≥ k + 1}

= inf{kRn Bn An − F k : dim ker F ≥ k + 1}

≤ inf{kRn Bn An − Rn Bn F k : dim ker F ≥ k + 1} ≤ kRn kkBn ksrk+1 (An ),

hence α(A) ≤ k + 1 exists. Analogously we find a β-number for A, that is, the third assertion holds. Finally, let A have an α-number and a β-number and let N ∈ N be such that inf{srα(A)+1 (An ), slβ(A)+1 (An ) : n ≥ N } > 0. Then An , n ≥ N , are Fredholm operators by Corollary 1.38. In view of Equation (1.11) there are a constant C > 0 and a sequence {Rn } ∈ F of projections Rn with kernels of dimension α(A) such that inf{kAn xk : x ∈ im Rn , kxk = 1} ≥ C for all n ≥ N . We consider the restrictions An |im Rn of An to im Rn which are injective. The spaces im(An |im Rn ) are of the codimension not greater than α(A) + β(A), hence they are closed and we can choose projections Sn onto im(An |im Rn ), which are uniformly bounded with respect to n ≥ N . Therefore, the operators (−1) An |im Rn : im Rn → im Sn are invertible and their inverses An are (uniformly) bounded by C. (−1) For the operators Bn := Rn An Sn we conclude that An Bn = An Rn A(−1) Sn = An A(−1) Sn = Sn , n n Bn An = Bn An Rn + Bn An (Ln − Rn ) = Rn A(−1) Sn An Rn + Bn An (Ln − Rn ) n

(2.7)

= Rn + Bn An (Ln − Rn ).

Since the latter term is of uniformly bounded rank, this proves the Fredholmness of A. Corollary 2.23. Let A = {An } ∈ F be a Fredholm sequence and let T , as well as Et , P t , (Ltn ) and (Ent ) be given as in Section 2.1 such that the Conditions (I), (II) are in force. Suppose that, for one t ∈ T , a subsequence of (Ent (An )Ltn ) converges P t -strongly to an operator At ∈ L(Et , P t ). Then At is Fredholm. If for all t in a certain subset T˜ ⊂ T and with respect to one common subsequence of A such operators At exist then X t∈T˜

dim ker At ≤ α(A),

X t∈T˜

dim coker At ≤ β(A).

Proof. The assertion immediately results from Theorem 2.9. For a setting F T and for sequences A ∈ F T we get from Theorem 2.13 Corollary 2.24. If A ∈ F T is a regular J T -Fredholm sequence then A is a Fredholm sequence in F, that is A is invertible modulo I. Moreover, the numbers α(A) and β(A) in the Definitions 2.10 and 2.20 are consistent.

2.4. RICH SEQUENCES

2.4

53

Rich sequences

Recall our motivating questions on band-dominated operators from Section 1.5 and let us consider two examples on lp (Z, C). • Firstly, set A := I + K with a P-compact operator K. Then (An ) converges to A whereas the two shifted copies (A1n ) and (A2n ) converge P-strongly to the identity. This can be easily seen, e.g. for (A1n ) one has A1n = Vn (Pn (I + K)Pn + Qn )V−n = Vn (I + Pn KPn )V−n = I + Vn Pn KPn V−n , and kPm Vn Pn Kk, kKPn V−n Pm k → 0 as n → ∞. We already promised (without having proved it yet!) that these snapshots are also sufficient to characterize the stability, that is we even claim that (An ) is stable if and only if A is invertible. • As a second example let J be given by the rule (xi ) 7→ ((−1)i xi ), that is its matrix representation equals diag(. . . , −1, 1, −1, 1, −1, 1, . . .), and set A := J+K with a P-compact K. The central snapshot is again A, but the P-strong limits of (A1n ) and (A2n ) do not exist anymore. To achieve convergence, we have to pass to subsequences, and this leads to two snapshots for (A1n ), namely P JP +Q and −P JP +Q, where P and Q are the complementary projections χZ+ I and χZ\Z+ I, respectively. Similarly, (A2n ) provides two further snapshots. We again announce that these five operators tell us everything about the stability. The pending proof will be given in Section 2.5.1, but for the moment we learned something very important: Instead of considering the whole sequences A in F one might pass to subsequences to attain a plainer structure. More precisely, one could fix a strictly increasing sequence g = (gn ) of positive integers and pass to the algebra Fg of all subsequences Ag = {Agn }. Obviously, the theory for elements in the respective algebras of subsequences Gg , FgT , and JgT is analogous to what we have already done for the basic case gn = n. We call a sequence of operators T -structured, if for every t ∈ T the respective snapshot W t (·) exists. Our second example teaches that the finite section sequences of band-dominated operators are not T -structured in general, but one could pass to subsequences which are T -structured, and hence satisfy the requirements for our Fredholm theory. Definition 2.25. We say that a sequence A ∈ F is uniformly rich with respect to T (or T -rich) if each strictly increasing sequence g of positive integers has a subsequence h ⊂ g such that Ah ∈ FhT .

To repeat this definition in other words, one may say that A ∈ F is T -rich if and only if every subsequence of A has a T -structured subsequence. Of course, one cannot expect the same behavior (like the existence of splitting numbers) under these weaker assumptions, but the following theorem shows that the stability, the general Fredholm property of Section 2.3 as well as the α- and β-numbers can still be well described in terms of snapshots. Theorem 2.26. . • The set RT of all T -rich sequences is a Banach algebra F T ⊂ RT ⊂ F. Moreover, the algebras RT and RT /G are inverse closed in F and F/G, respectively.

• If every T -structured subsequence of A ∈ RT has a regularly JhT -Fredholm subsequence then A is Fredholm and " # " # X X t t α(A) = max dim ker W (Ag ) , β(A) = max dim coker W (Ag ) , (2.8) t∈T

t∈T

54


where the maximum is taken over all (T -structured) subsequences Ag ∈ FgT of A. Moreover, in this case, for every (T -structured) subsequence Ag ∈ FgT of A the numbers α(Ag ) and β(Ag ) are splitting numbers and X lim ind Agn = ind W t (Ag ). n→∞

t∈T

• For A ∈ RT the following are equivalent: 1. 2. 3. 4. 5. 6. 7.

A is stable. Every subsequence of A is stable. Every subsequence of A has a stable subsequence. Every T -structured subsequence of A is stable. Every T -structured subsequence of A has a stable subsequence. Every T -structured subsequence of A is JhT -Fredholm and all snapshots are invertible. Every T -structured subsequence of A has a JhT -Fredholm subsequence and all snapshots are invertible.

Proof. The proof of RT being a closed algebra F T ⊂ RT ⊂ F is straightforward. For the second assertion assume that A = {An } does not have an α-number. Then for each n ∈ N there is a number jn such that jn > jn−1 and srn (Ajn ) < 1/n. Choose a JhT -Fredholm subsequence Ah ∈ FhT of Aj and find a splitting number for (srk (Ahn )) due to Theorem 2.13, a contradiction. Thus, A has an α-number and, analogously, a β-number, and Theorem 2.22 yields the Fredholm property of A. From Theorem 2.9 or Corollary 2.23 we deduce the relations “≥” in (2.8). Furthermore, there is a sequence j tending to infinity such that limn slα(A) (Ajn ) = 0. Choose a subsequence Ag ∈ FgT of Aj and pass to the JhT -Fredholm subsequence Ah . Notice that Ag and Ah have the same snapshots. Then Theorem 2.13 applies to Ah and yields the equality for α(A) in (2.8). Analogously we proceed for β(A). Now, let Ag be a T -structured subsequence of the Fredholm sequence A. Then α(Ag ) < ∞. Assume that α(Ag ) is not a splitting number. Then there is a subsequence Ah of Ag such that limn srα(Ag ) (Ahn ) exists and is strictly positive, hence α(Ah ) < α(Ag ). Pass to a JjT -Fredholm subsequence Aj of Ah which certainly has the splitting property with α(Aj ) being the sum of the kernel dimensions of the snapshots, a contradiction. For the β(Ag ) and the index formula we proceed in an analogous manner. We now turn our attention to the third assertion. If a sequence is stable then every subsequence is so, and it is obvious that 1. ⇒ 2. ⇒ 3. ⇒ 5. and 1. ⇒ 2. ⇒ 4. ⇒ 5.. Furthermore, 4. ⇔ 6. and 5. ⇔ 7. hold (see Theorem 2.17). Statement 7. implies that A is Fredholm with α(A) = β(A) = 0 by the second assertion. Thus, Theorem 2.21 yields the stability of A, i.e. 7. ⇒ 1. Finally, let B be a G-regularizer for A and g be a sequence tending to infinity. We pass to a subsequence h ⊂ g with Ah ∈ FhT . Then Ah + Gh is invertible in FhT /Gh by Theorem 2.17 and the (unique) inverse is given by Bh + Gh , hence Bh ∈ FhT . Since g was chosen arbitrarily, this shows that B ∈ RT and finishes the proof of the first assertion. Remark 2.27. We note that a T -structured subsequence Ag of a Fredholm sequence A is not necessarily JgT -Fredholm. To see this simply take a setting where T is not a singleton, fix t ∈ T and a P t -compact finite rank projection K, set T˜ := T \ {t} and consider the sequence ˜ A = {An } := I − {Ent (Ltn KLtn )}. Clearly, A is Fredholm and belongs to F T , but there is no T˜ subsequence Ag being Jg -Fredholm. Moreover, the sequence B = {Bn }, given by Bn = An if n is even and Bn = Ln otherwise, is Fredholm and T˜-structured, but has no splitting numbers.

2.5. EXAMPLE: FINITE SECTIONS OF BAND-DOMINATED OPERATORS

2.5

55

Example: finite sections of band-dominated operators

Let us take another look at the band-dominated operators in Alp . We now give a precise definition of the sequence-algebraic framework for the finite section sequences of band-dominated operators.

2.5.1

Standard finite sections

Let H+ stand for the set of all strictly increasing sequences of positive integers and, analogously, H− for the set of all strictly decreasing sequences of negative integers. For a given sequence h = (hn ) ∈ H+ we set Lhn := Phn , Ehn := im Lhn , T := {−1, 0, +1}, I 0 := I, I ±1 := χZ∓ I and E0 := lp (Z, X)

L0hn := Lhn Eh0n : L(im L0hn ) → L(Ehn ), B 7→ B

E±1 := im I ±1

L±1 hn := V∓hn Lhn V±hn

Eh±1 : L(im L±1 hn ) → L(Ehn ), B 7→ V±hn BV∓hn n for every n. By P t := (Ltn ) uniform approximate identities on Et are given such that Et have the P t -dichotomy, and the sequences (Lthn ) converge P t -strongly to the identities I t on Et . As usual, we let FhT denote the Banach algebra of all bounded sequences {Ahn } of bounded linear operators Ahn ∈ L(Ehn ) for which there exist operators W t {Ahn } ∈ L(Et , P t ) for each t ∈ T such that for n → ∞ Eh−t (Ahn )Lthn → W t {Ahn } n

P t -strongly.

Further, we introduce the closed ideals Gh and JhT in FhT by Gh := {{Ghn } : kGhn k → 0},

JhT := span{{Eht n (Lthn KLthn )}, {Ghn } : t ∈ T, K ∈ Kt , {Ghn } ∈ Gh }. A sequence {Ahn } ∈ FhT is said to be JhT -Fredholm, if {Ahn } + JhT is invertible in FhT /JhT . The finite section algebra FAlp Let F stand for the algebra of all bounded sequences {An } of bounded linear operators An ∈ L(En ) and let FAlp denote the smallest closed subalgebra of F containing all sequences {Ln ALn } with some rich A ∈ Alp . For A = {An } ∈ FAlp and a sequence h = (hn )n∈N ∈ H+ let Ah denote the subsequence {Ahn }. It is obvious, that for all A = {An } ∈ FAlp and each h ∈ H+ the central snapshot exists independently from the choice of h: W (A) := P-lim An Ln = W 0 (Ah ) = P-lim Ahn Lhn . n→∞

n→∞

Moreover, for A = {Ln ALn }, A ∈ Alp being rich, there is a sequence g ∈ H+ such that Ag exists, and further we can choose a subsequence h ⊂ g which also provides A−h , besides Ah = Ag . Thus A ∈ RT , and since RT is a closed algebra of F we even find that FAlp ⊂ RT . The following proposition is the key for the application of the Fredholm theory, which was introduced in the preceding sections, to the algebra Alp .

56


Proposition 2.28. Let A ∈ FAlp be such that W (A) is P-Fredholm and let g ∈ H+ . Then there is a T -structured subsequence Ah of Ag being JhT -Fredholm. Proof. We initially check that, for every j ∈ H+ and for each pair of operators B, C ∈ Alp such that B±j and C±j exist, we have D := {Ljn BCLjn } − {Ljn BLjn }{Ljn CLjn } ∈ JjT .

(2.9)

m m m T For this, put Qm j := {Ljn −m } and Pj := Ij − Qj and check that Pj belongs to Jj . Since the T m limit operators B±j , C±j exist, we have D ∈ Fj . For every m the sequence DPj is contained in JjT and kDQm j k ≤ kBk sup k(I − Ljn )CLjn −m k n∈N

which can be chosen arbitrarily small by Theorem 1.55. Thus, the closedness of JjT implies (2.9). We now show that Ah − {Lhn W (A)Lhn } ∈ JhT (2.10) for a certain subsequence h ⊂ g. Note that there is a sequence (of sequences) (A(m) ) ⊂ FAlp such that A(m) → A in the norm of FAlp as n → ∞, and such that all A(m) are of the form A(m) =

km Y lm X

i=1 j=1

(m)

{Ln Aij Ln },

(m)

Aij

∈ Alp . (1)

We choose a subsequence g 1 of g (applying (2.9)) such that all limit operators of all Aij exist (1)

and such that (2.10) holds for Ag1 . Repeating this argument we can obtain a subsequence g 2 (2)

(2)

of g 1 such that all limits of all Aij exist and such that (2.10) holds for Ag2 as well, and so on. (k)

Hence, there are sequences g ⊃ g 1 ⊃ g 2 ⊃ . . . ⊃ g m ⊃ . . . such that Agm ∈ FgTm and (2.10) holds (k)

for Agm for every m and k ≤ m. Now let the sequence h = (hn ) be defined by hn := gnn . Since (m)

Ah is the norm limit of the sequence (Ah )m∈N , we easily obtain (2.10) for Ah . Finally, let B be a P-regularizer for A := W (A) and note that B ∈ Alp by Theorem 1.55. The construction of h provides the limit operators A±h , and the proof of Proposition 1.59 shows that B±h exist as well, thus we can apply (2.9) to the operators A and B. This yields the JhT -Fredholmness of {Lhn W (A)Lhn } and hence also the JhT -Fredholmness of Ah , by (2.10).

Definition 2.29. Let HA ⊂ H+ stand for the set of all sequences h such that Ah ∈ FhT . Corollary 2.30. Let A ∈ FAlp and W (A) be P-Fredholm. Then W ±1 (Ag ) are P ±1 -Fredholm for every g ∈ HA . Proof. Choose a subsequence h ⊂ g such that Ah is JhT -Fredholm by the previous proposition. Its snapshots W t (Ah ) equal W t (Ag ) and are P t -Fredholm, respectively (see Theorem 2.6). The main results Now we are in a position to adapt the general results of Part 2 to this specific class of approximation sequences of band-dominated operators and we are going describe the Fredholm properties of both, the T -structured subsequences of A ∈ FAlp as well as the whole sequence A.


57

Theorem 2.31. Let A = {An } ∈ FAlp and g ∈ HA . • If W (A), W ±1 (Ag ) are Fredholm, then lim ind Agn = ind W (A) + ind W +1 (Ag ) + ind W −1 (Ag )

n→∞

and the approximation numbers from the right/left of the operators Agn of Ag have the α-/β-splitting property with α = dim ker W (A) + dim ker W +1 (Ag ) + dim ker W −1 (Ag ), β = dim coker W (A) + dim coker W +1 (Ag ) + dim coker W −1 (Ag ). • If one of the operators W (A), W ±1 (Ag ) is not Fredholm then, for each k ∈ N, we have limn srk (Agn ) = 0 or limn slk (Agn ) = 0. • Ag is stable if and only if W (A) and W ±1 (Ag ) are invertible. Proof. Let all snapshots of Ag be Fredholm, pass to a subsequence Ah of Ag which is regularly JhT -Fredholm, by Proposition 2.28, and apply Theorem 2.26 to prove the first assertion. The second one easily follows from Theorem 2.9, and the last one again from Theorem 2.26 and Proposition 2.28. For the whole sequence A ∈ FAlp we further conclude from Theorem 2.26, Proposition 2.28 and Corollary 2.23 Theorem 2.32. Let A = {An } ∈ FAlp . Then, A is a Fredholm sequence if and only if W (A) and all operators W ±1 (Ah ) with h ∈ HA are Fredholm. In this case the α- and β-number of A equal α(A) = dim ker W (A) + max dim ker W +1 (Ah ) + dim ker W −1 (Ah ) , h∈HA (2.11) β(A) = dim coker W (A) + max dim coker W +1 (Ah ) + dim coker W −1 (Ah ) . h∈HA

A sequence A ∈ FAlp is stable if and only if all snapshots W (A) and W ±1 (Ah ) with h ∈ HA are invertible.

Notice that every P-Fredholm operator is Fredholm if dim X < ∞. Thus, the last theorem, together with Corollary 2.30 reveal that the Fredholm property of W (A) is already sufficient for the Fredholm property of A in this case. The stability criterion of Theorem 2.32 often appears in a slightly different form in the literature and we want to close this paragraph with an outline of this alternative notation. For A = {An } in FAlp denote the set of all operators Bh which are P-strong limits of one of the sequences (V−hn [(I − Lhn ) + Lhn Ahn Lhn ]Vhn ) with h ∈ H+ (or H− ) + − + − by σstab (A) (or σstab (A), respectively). Of course, if Bh is in σstab (A) or σstab (A) then we can pass to a subsequence g ∈ HA of h and identify Bh with the respective operator W ±1 (Ag ).

Theorem 2.33. A sequence A ∈ FAlp is stable if and only if all operators in + − σstab (A) := {W (A)} ∪ σstab (A) ∪ σstab (A)

are invertible.

58


Remark 2.34. This result was recently proved in [43] for the finite section sequence (Ln ALn ) of a single band-dominated operator A, also by studying its subsequences. It has a lot of predecessors, which required additional restrictions like p = 2, 1 < p < ∞, a uniform bound for the norms of the inverses of all operators in σstab {Ln ALn }, or the existence of a predual setting (see [61], [64], [45], [44], [15], for example). Now having this result for sequences in the whole algebra FAlp , we get much more flexibility in constructing efficient algorithms for specific operators. Assume, for example, that the operator A admits a decomposition m Y k X A= Aij i=1 j=1

into band-dominated operators of simple structure (e.g. banded, triangular, or Toeplitz operators). This structure, which could permit the application of fast algorithms, gets lost if one applies the usual finite section method Ln ALn , but it can be preserved, if one uses the composition of the single finite sections Ln Aij Ln instead. The results above provide the desired information on the stability and convergence also for such compositions.

2.5.2

Adapted finite sections

In the preceding section the aim was to check if for arbitrarily given band-dominated operators A the “standard” finite section method, based on the projections Pn , applies. Providing that A is rich, the answer is that one has to check the Fredholm properties and the invertibility of a possibly infinite set σstab (A) of operators. Of course, we have seen that we can pass to subsequences to reduce the number of limit operators, but since the projections Ln are always chosen symmetric w.r.t. Z, the limiting processes towards ∞ and −∞ are somehow coupled, which seems to be artificial. Now we let A ∈ Alp be fixed and we ask if there is a more adapted sequence of projections which provides a specific “finite section like” method for this single operator, such that we only have to check one snapshot at ∞ and one snapshot at −∞ and such that both can be chosen independently from each other. The idea is very natural and simple: Suppose that for A ∈ Alp (not necessarily rich) there is a sequence l ∈ H− , such that Al exists and, independently of l, let u ∈ H+ be another sequence such that Au exists. We show that the properties of the finite l,u sections {Ll,u n ALn }, with Ll,u n = χ{l(n),...,u(n)} I, are determined by the three operators A, Al and Au . Theorem 2.35. Let A ∈ Alp , and let l ∈ H− , u ∈ H+ be such that Al , Au ∈ σop (A) exist. l,u l,u l,u l,u Further let A := {Al,u n } denote the sequence of the operators An := Ln ALn ∈ L(im Ln ). Then • The operators A+ := χZ+ Al χZ+ I + (1 − χZ+ )I and A− := χZ− Au χZ− I + (1 − χZ− )I are P-Fredholm, whenever A is P-Fredholm. + − • If A and A± are Fredholm then limn ind Al,u and n exists, equals ind A + ind A + ind A l,u the approximation numbers from the right/left of An have the α-/β-splitting property with α = dim ker A + dim ker A+ + dim ker A− , and β w.r.t. cokernels instead of kernels.

• If one of the operators A, A± is not Fredholm then, for each k ∈ N, limn srk (Al,u n ) = 0 or limn slk (Al,u ) = 0. n • A is stable if and only if A and A± are invertible.


59

Proof. Notice that (Ll,u n ) converges P-strongly to the identity and introduce T := {−1, 0, +1}, l,u,T homomorphisms Ent : L(im Ll,u,t ) → L(El,u , J l,u,T in the same n n ) and sequence algebras F way as in Section 2.5.1. Then the finite section sequence of each P-regularizer of A is again in F l,u,T , is a J l,u,T -regularizer for A (cf. (2.9)) and Theorems 2.6, 2.13, 2.14, 2.9, and 2.17 give the claim. Of course, these two slightly different approaches can be combined to get more appropriate methods for classes of band-dominated operators having a certain common structure: First, one chooses the sequence (Ll,u n ) of projections which align with the common structure in a sense and one then proceeds as in Section 2.5.1 to prove Theorems 2.31 and 2.32 also in this setting. Furthermore, [79] and [80] present an idea how one can pass to modified finite sections which need not to be stable, but which are still generalized invertible (e.g. Moore-Penrose invertible). Also note that the equations Ax = y and Vκ Ax = Vκ y are equivalent for every κ ∈ Z since Vκ is invertible, whereas at most one of them leads to a stable finite section sequence and therefore can be solved by the finite section method. The simplest example of an operator for which such a preconditioning procedure is indicated to get a stable finite section sequence is the operator A = V1 itself. This method is known as index cancellation and was already studied in [24]. A comprehensive discussion can also be found in [36]. A final remark The core of the simplifications which appear in this more specific situation of finite sections of band-dominated operators is given by Proposition 2.28: The Fredholm properties of a sequence A are already determined by the Fredholm properties of the central snapshot W (A), and hence the “J.T -Fredholmness” can be extinguished from all results. This may look nice and simple for the moment, but it is still unsatisfactory in a sense, because it can hardly be translated to other classes of operators or lifted to a more abstract level. Therefore, the next part is devoted to this gap, and the main tools there are localization techniques. This deeper study will provide further amazing relations between the sequences and their snapshots, for instance on the asymptotics of condition numbers and pseudospectra.

60


Part 3

Banach algebras and local principles In the previous part we have seen how the stability and invertibility problem in certain sequence algebras can be replaced by the invertibility problem for a family of operators (the snapshots) and for a coset in a smaller quotient algebra (namely F T /J T ). Of course, the latter condition is still unpleasant and can hardly be checked directly. Therefore it would be desirable to get a family of snapshots which is sufficient in the sense that the Fredholm property or invertibility of the snapshots can already provide the invertibility of the coset in question. It is not clear if or how this could be achieved in the very general context F T . So, here we draw on well known local principles to attack this question for classes of sequences which will be referred to as localizable sequences in what follows. For this we start with a short introduction and description of the required tools in Section 3.1 and proceed with their application to sequence algebras in 3.2. We achieve our goal, the characterization of the Fredholm property of localizable sequences solely in terms of its snapshots, with Theorem 3.9. Furthermore, we consider the asymptotic behavior of norms, condition numbers and pseudospectra. This approach was already evolved and published for the finite sections of band-dominated operators in [82], but is new in this abstract framework which permits to consider further applications in Part 4. The third section is devoted to the pending proofs in the theory on approximate projections. Here we look at this concept from a new, more general and purely Banach algebraic point of view.

62

PART 3. BANACH ALGEBRAS AND LOCAL PRINCIPLES

3.1

Central subalgebras, localization and the KMS property

Classical Gelfand theory 1 A mapping ∗ : A 7→ A∗ on a Banach algebra A is referred to as an involution if for all A, B ∈ A and all scalars α, β ∈ C (A∗ )∗ = A,

¯ ∗ (αA + βB)∗ = α ¯ A∗ + βB

and (AB)∗ = B ∗ A∗ .

We say that the unital Banach algebra A with the involution ∗ is a C ∗ -algebra if for every A ∈ A the equality kA∗ Ak = kAk2 holds true. Note that involutions on C ∗ -algebras A are isometries, that is kA∗ k = kAk for every A ∈ A. Further, a Banach algebra C is said to be commutative, if AB = BA holds for all A, B ∈ C.

So, in what follows let C be a commutative C ∗ -algebra. A character of C is a non-zero homomorphism from C into the algebra C. We denote the set of all characters by MC and mention that every character is unital with norm 1. Every element A ∈ C defines a complex-valued function ˆ Aˆ on MC , its so-called Gelfand transform, by the rule A(h) := h(A). The coarsest topology on MC which makes every Aˆ continuous is referred to as the Gelfand topology. An ideal I ⊂ C is said to be maximal, if the only ideal of C which is a proper superset of I is C itself. Theorem 3.1. (Gelfand-Naimark Theorem) Let C be a commutative C ∗ -algebra. Then the mapping h 7→ ker h is a bijection from the set MC of the characters h of C onto the set of the maximal ideals of C. Provided with the Gelfand topology, MC becomes a compact Hausdorff space and the Gelfand transformation A 7→ Aˆ is an isometric isomorphism from C onto C(MC ), the algebra of all continuous functions on MC . Thus, one can think of the elements h of MC either as characters on C or as maximal ideals of C. In the literature both characterizations are often used synonymously, and one usually refers to MC as the maximal ideal space of C. Due to this theorem, C can be isometrically identified with the algebra of the continuous functions on the maximal ideal space MC . Clearly, an element A ∈ C is invertible if and only if its Gelfand transform Aˆ does not have any zeros on MC (one may say that it is locally invertible in every ˆ point). Furthermore the norm of an element A equals the norm of its Gelfand transform A. Since the latter is a continuous function on the compact set MC , we have ˆ kAk = max |A(h)|. h∈MC

(3.1)

Therefore, in all what follows, the elements of MC are denoted by x and are called points, the elements of C are denoted by ϕ and are called functions, as a rule, and we will be a bit loosely and use the notation ϕ(x) for ϕ(x). ˆ For the identification of the maximal ideal space MC one often benefits from the Banach-Stone theorem, whose proof can be found in [21], V.8.8. Theorem 3.2. Let H, K be compact Hausdorff spaces, and let T be an isometric isomorphism between C(H) and C(K). Then H and K are homeomorphic, that is there is a continuous bijection τ : H → K whose inverse is continuous as well. In the case of non-commutative Banach algebras the situation is much more involved and we here mainly build upon the paper [10] by Böttcher, Krupnik and Silbermann. 1 These

results and their proofs can be found in every textbook on C ∗ -algebras (e.g. [30], Section 4.1).

3.1. CENTRAL SUBALGEBRAS, LOCALIZATION AND THE KMS PROPERTY

63

Local principles for non-commutative Banach algebras Let A be a unital Banach algebra. The center cen A of A is the set of all elements C ∈ A with the property that CA = AC for all A ∈ A. Let C be a C ∗ -subalgebra of cen A containing the identity. For each point x in the maximal ideal space MC we introduce Jx , the smallest closed ideal in A containing x,2 and let φx denote the canonical mapping from A to A/Jx . In case Jx = A the algebra A/Jx consists of only one element Θ which is zero and the identity at the same time, and therefore we may say that it is invertible and has norm 0. Theorem 3.3. (Allan [1], Douglas [19], see also [12], Theorem 1.35 or [63], Theorem 2.3.16f ) The element A ∈ A is invertible if and only if φx (A) is invertible in A/Jx for every x ∈ MC . The mapping MC → R+ , x 7→ kφx (A)k is upper semi-continuous. If φx (A) is invertible then there is a neighborhood U of x in MC such that φy (A) is invertible in A/Jy for every y ∈ U .

This again provides a “local characterization” of invertibility in A. We are also interested in a formula which represents the norm P of an element A in terms of local norms as in (3.1). Notice that the set of all finite sums i ϕi Ai with some ϕi ∈ C, ϕi (x) = 0 and Ai ∈ A forms a dense subset of Jx , and, by Proposition 5.1 of [10], kφx (A)k = inf{kϕAk : ϕ ∈ C, 0 ≤ ϕ ≤ 1, ϕ ≡ 1 in a neighborhood of x}.

(3.2)

∗

Definition 3.4. Let A be a unital Banach algebra and C a central C -subalgebra which contains the identity. The algebra A is a KMS-algebra with respect to C 3 if for every A ∈ A and ϕ, ψ ∈ C with disjoint supports (i.e. ϕ(x)ψ(x) = 0 for every x) k(ϕ + ψ)Ak ≤ max{kϕAk, kψAk}.

(3.3)

Theorem 3.5. (Böttcher, Krupnik, Silbermann, [10], Theorem 5.3) The algebra A is a KMS-algebra with respect to C if and only if for every A ∈ A kAk = max kφx (A)k. x∈MC

(3.4)

A look into the proof of this Theorem in [10] reveals that one only has to check for every single A (separately) that (3.3) holds for all ϕ, ψ ∈ C with disjoint supports if and only if (3.4) is true. Thus, we can flexibilize the notion of KMS-algebras a little bit without any change in Theorem 3.5: Definition 3.6. Let B be a unital Banach algebra and C a central C ∗ -subalgebra containing the identity. A subalgebra A ⊂ B is said to be a KMS-algebra with respect to C if (3.3) holds for every A ∈ A and ϕ, ψ ∈ C with disjoint supports. The aim of the following considerations is to use this concept of localization to find practicable criteria for the J T -Fredholm property of a structured operator sequence in F T as introduced in Part 2. To be a bit more precise, we want to characterize a class of settings F T where the P t Fredholm property of the snapshots implies “local J T -Fredholmness” of the sequence and hence its J T -Fredholmness in the following sense: The local principle of Allan and Douglas suggests to look for a suitable C ∗ -subalgebra C and to replace the J T -Fredholmness by the invertibility of all respective local cosets. As a second step one then might try to identify these local cosets with the snapshots to get the desired result. Both the existence of C and the identification are not achievable in the very general context of sequence algebras F T , but under additional conditions. In what follows we try to give a framework which is sufficiently restrictive to manage these two problems, but on the other hand, also sufficiently general to cover the algebras of more concrete operator sequences which we have in mind and which we take into the focus in the 4th part of this thesis. precisely, Jx is the smallest closed ideal in A containing the kernel of the character x : C → C. recent literature (e.g. [72]) such pairs (A, C) are also called faithful localizing pairs.

2 More 3 In

64


3.2

Localization in sequence algebras

Let F T be a sequence algebra as in Section 2.1 et seq.

3.2.1

Localizable sequences

A localizing setting Let C ⊂ F T be a closed algebra containing I such that every snapshot W t (ϕ) of every sequence ϕ ∈ C is just a multiple ϕ(t)I t of the respective identity, and such that for t1 , t2 ∈ T , t1 6= t2 there is always a ϕ ∈ C with ϕ(t1 ) 6= ϕ(t2 ). Further suppose that C/G := {ϕ + G : ϕ ∈ C} can be provided with an involution which turns C/G into a commutative C ∗ -algebra. From Theorem 3.1 we know that C/G can be identified with the set of all continuous functions on its maximal ideal space MC/G . Notice that, for every t ∈ T , the mapping C/G → C which sends ϕ + G to ϕ(t) is a character, that means T can be understood as a subset of MC/G . Let M be a set T ⊂ M ⊂ MC/G such that the closure of MC/G \ M is either empty or intersects M . For every x ∈ M \ T we also fix one “direction” t = t(x) ∈ T and say that it is associated to x. In this way the objects Ex := Et(x) , and analogously P x , Enx , Lxn , etc., and consequently the snapshots W x are defined at every point x ∈ M . ˜ x ∈ C(MC/G ) (that is F ˜ x ∈ C/G) which takes values Finally, for each x ∈ M , fix one function F x ˜ x , and set ˜ in [0, 1] such that F ≡ 1 in a neighborhood of x, choose one sequence Fx = (Fnx ) ∈ F x x −x x ˜ x may be (Sn ) := (En (Fn )Ln ). Now, we say that C defines a localizing setting. The functions F regarded as “local filter” in x, respectively. Having such a localizing setting C we are going to single out a family of sequences A ∈ F T which can be studied using localization with respect to C, and whose local representatives are already characterized by the snapshots W t (A). Localizable sequences Suppose that C defines a localizing setting. A sequence A ∈ F T is said to be C-localizable if, for every x ∈ M , • W x (A) is liftable, that is Ax := {Enx (Snx W x (A)Snx )} = Fx {Enx (Lxn W x (A))}Fx ∈ F T ,

• W x (A) is almost commuting with Fx , that is k[W x (A), Snx ]k → 0 as n → ∞,

• A + G and Ax + G belong to the commutant of C/G in F/G, that means they commute with every coset ϕ + G, ϕ ∈ C,

• A + G and Ax + G are locally equivalent in x, that is for every > 0 there is a function ϕ + G ∈ C/G being equal to 1 in a neighborhood of x such that kϕ(A − Ax ) + Gk < .

The set of all C-localizable sequences shall be denoted by LT = LT (C). Finally, suppose that there is a sequence T = {Tn } ∈ LT of projections such that T + G is locally equivalent to zero in every x ∈ MC/G \ M , and let T T denote the set of all “truncated” sequences of the form TAT + λ(I − T) + J with A ∈ LT , a complex number λ and a sequence J ∈ J T . Remark 3.7. Of course, it is possible (and even the rather common case) to choose M = MC/G and T = I. Then also T T = LT holds, as we obtain from the next proposition. In this case we say that C defines a globally localizing setting. Otherwise C is said to define a partially localizing setting. Notice that this model will enable us to consider sequence algebras which contain the constant sequences A = {A} of the operators of interest as well as the sequence T of projections which constitutes the “finite section method”, and to study the finite section sequences of the form TAT + (I − T). For an application of this kind see Section 4.3.

3.2. LOCALIZATION IN SEQUENCE ALGEBRAS

65

Proposition 3.8. The sets T T and LT are Banach algebras, fulfill T T ⊂ LT and contain G as well as J T as closed ideals. The set C/J T := {ϕ + J T : ϕ ∈ C} forms a commutative C ∗ -subalgebra of F T /J T being ∗-isomorphic to C/G and C(MC/G ). Proof. Obviously, LT is a closed linear subspace of F T . For A, B ∈ LT and x ∈ M we have that W x (AB) is almost commuting with Fx , AB + G belongs to the commutant of C/G, Ax Bx ∈ F T and Fx (AB)x Fx − Ax Bx = {Enx (Snx Snx W x (AB)Snx Snx − Snx W x (A)Snx Snx W x (B)Snx )} ∈ G since W x (A) and W x (B) are almost commuting with Fx , hence Fx (AB)x Fx ∈ F T , too. For every t ∈ T with W t (Fx ) = 0 we have W t ((AB)x ) = 0, and for all other t we know that the snapshot W t (Fx ) is a non-zero multiple dI t of the identity, hence (En−t ((AB)xn )Ltn ) converges P t -strongly if and only if (En−t (Fnx (AB)xn Fnx )Ltn ) converges P t -strongly. The only if is trivial, and for the if part let K be P t -compact. Then it holds that 4 d2 KEn−t ((AB)xn )Ltn =G KEn−t ((Fnx )2 (AB)xn )Ltn =G KEn−t (Fnx (AB)xn Fnx )Ltn , as well as the dual equation with K multiplied from the right. This shows that (AB)x ∈ F T , i.e. W x (AB) is liftable. Moreover, (AB)x + G belongs to the commutant of C/G since x x x x x ϕ(AB)x = ϕ{Enx (Snx W x (AB)Snx )} =G {Enx (ϕ(x) n Sn W (A)Sn W (B)Ln )}

x x x x x (x) x x =G {Enx (Snx W x (A)Snx ϕ(x) n W (B)Ln )} =G {En (Sn W (A)ϕn W (B)Sn )}

x x x x x x x x x (x) =G {Enx (Lxn W x (A)ϕ(x) n Sn W (B)Sn )} =G {En (Ln W (A)Sn W (B)Sn ϕn )}

(3.5)

=G {Enx (Snx W x (AB)Snx )}ϕ = (AB)x ϕ

for every ϕ ∈ C. These equalities also give that Ax Bx + G and (AB)x + G are locally equivalent in x 5 The proof of LT being a Banach algebra can now easily be completed. Furthermore, T T clearly forms an algebra since T2 = T and J T is a closed ideal in F T , and a simple approximation argument yields its closedness. The set G is a closed ideal in both algebras T T and T T , as well as J T in T T . To prove this for J T in LT , it suffices to consider the generating sequences A = (Enτ (Lτn K)) with some τ ∈ T and K ∈ K(Eτ , P τ ). Its snapshots are P x -compact, respectively, and hence are liftable and almost commuting with Fx . The membership of A to the commutant of C/G follows from the fact that the snapshot W τ (·) of every C-sequence is a multiple of the identity. To prove the local equivalence of A to the respective lifting notice that A and Ax are locally equivalent (they even coincide) in every x ∈ M with t(x) = τ , since W x (A) = K in these cases. Let x ∈ M be such that t(x) 6= τ . Then W x (A) = 0. Choose a function ϕ ∈ C which is equal to 1 in a neighborhood of τ but vanishes in a neighborhood of x. Then ϕA − A ∈ G and ϕA is locally equivalent to zero in x. Hence A is locally equivalent to zero in x as well. Now the assertion easily follows for all J T -sequences. Furthermore, the inclusions T T ⊂ LT ⊂ F T are proved. Clearly, C/J T is a commutative subalgebra of F T /J T and it inherits its involution from C/G. It remains to show that kϕ + Gk = kϕ + J T k always holds. The estimate “≥” is obvious. Assume that it is even proper for one ϕ, which means that there is an > 0 and a sequence J ∈ J T such that kϕ + Gk > kϕ + Jk + ≥ kϕ + J + Gk + . Without loss of generality we can also assume that kϕ + Gk = 1. Fix x0 ∈ MC/G such that d := (ϕ + G)(x0 ) fulfills |d| ≥ 1 − and x0 either belongs 4 Write

an =G bn if kan − bn k → 0 as n → ∞. choose ϕ such that ϕ(I − Fx ) ∈ G.

5 simply

66


to T or to the interior of MC/G \ T . In the latter case we choose a function ψ + G ∈ C/G equal to 1 in a neighborhood of x0 , equal to zero on T and of norm one. In case x0 ∈ T we set ψ := I. Then B + G := dI − ψ(ϕ + J) + G belongs to F T /G and is invertible with its inverse given by a Neumann series. Thus, all of its snapshots are invertible. In case x0 ∈ T this contradicts the fact that W x0 (B) is P x0 -compact. If x0 ∈ / T then ψJ ∈ G, hence W x0 (B) is zero, a contradiction as well. Theorem 3.9. The algebras LT , LT /G and LT /J T are inverse closed in F, F/G and F T /J T , respectively. The same holds true with LT replaced by T T . A sequence A ∈ T T is J T -Fredholm if and only if all of its snapshots are P t -Fredholm. It is Fredholm if and only if all of its snapshots are Fredholm. It is stable if and only if all of its snapshots are invertible. Proof. Let Com(C/G) ⊂ F/G denote the commutant of C/G, that is the set of all elements in F/G which commute with every element in C/G. It is well known that Com(C/G) ⊂ F/G is a closed subalgebra of F/G, which contains LT /G by definition. Furthermore, recall the homomorphisms φx from Section 3.1 which act on Com(C/G) and notice that two cosets A + G, B + G ∈ Com(C/G) are locally equivalent in x ∈ M if and only if φx (A + G) = φx (B + G). Indeed, the only if follows from the fact that φx (ϕ + G) = φx (I + G) for every ϕ ∈ C being equal to 1 in a neighborhood of x. For the if part P let φx (A − B + G) = 0. Then A − B + G can be approximated by finite sums of the form i ϕi Ci + G with some ϕi ∈ C, ϕi (x) = 0 and Ci ∈ F as mentioned after Theorem 3.3, and the assertion follows. The respective local homomorphisms on the commutant Com(C/J T ) ⊂ F T /J T of C/J T shall be denoted by Φx . Let A ∈ LT be J T -Fredholm and B ∈ F T one of its regularizers. We want to show that B ∈ LT . Let x ∈ M and consider the snapshot B := W x (B) which is a P x -regularizer for A := W x (A) by Theorem 2.6. We first prove that limn k[B, Snx ]k = 0 and that B is liftable. We again write an =G bn if limn kan − bn k = 0, and since I − AB, I − BA are P x -compact we have B(I x − Snx ) =G B(I x − Snx )AB =G BA(I x − Snx )B =G (I x − Snx )B. The snapshot W x (Bx ) of the sequence Bx := {Enx (Snx BSnx )} at the point x obviously exists, and for those t ∈ T with W t (Fx ) = 0 the snapshots W t (Bx ) exist as well and equal zero. Let t ∈ T \ {t(x)} be such that W t (Fx ) (which is a multiple dI t of the identity) does not vanish, and set At := W t (Ax ) = W t {Enx (Snx ASnx )}. For every K ∈ K(Et , P t ), kEn−t (Enx (Snx B(Snx )2 ASnx ))Kk kEn−t (Enx (Snx BSnx ))k 1 ≥ t x kEn−t (Enx (Snx BA(Snx )3 ))K + Gn k, M M sup kSnx k2 kBk

kEn−t (Enx (Snx ASnx ))Kk ≥

n∈N

where Gn := En−t (Enx (Snx B[A, (Snx )2 ]Snx ))K tend to zero in the norm as n → ∞. Thus, letting n go to infinity, we find kAt Kk ≥ ckKk with a constant c > 0 independent of K, which shows that At is injective, due to its P t -dichotomy. Analogously, one checks that At is surjective, that is (At )−1 exists, and it belongs to L(Et , P t ) by Theorem 1.14. We even have that Hn := d4 I t − En−t (Enx (Snx BSnx ))En−t (Enx (Snx ASnx ))Ltn converges P t -strongly to zero, hence, for every P t -compact K, k[En−t (Enx (Snx BSnx ))Ltn − d4 (At )−1 ]Kk

= kEn−t (Enx (Snx BSnx ))Ltn [At − En−t (Enx (Snx ASnx ))Ltn ](At )−1 K + Hn (At )−1 Kk


67

tends to zero as n → ∞. The dual relation with K multiplied from the left-hand side holds as well, thus the snapshot W t (Bx ) exists and equals d4 (At )−1 . We conclude that B is liftable. In the same way as in the equations (3.5) and with the help of the fact B − BAB ∈ K(Ex , P x ) we find that Bx + G is in the commutant of C/G. By B − BAB ∈ J T and ϕBAB =G BAϕB =G BϕAB =G BABϕ we see that B + G is in the commutant of C/G as well. Hence it remains to check that B and Bx are locally equivalent. Since the coset Φx (B − Bx + J T ) = Φx (BA(B − Bx ) + J T ) = Φx (B(AB − ABx ) + J T ) = Φx (B(AB − Ax Bx ) + J T ) = Φx (B(I − I) + J T )

equals zero, we have φx (B − Bx + G) ∈ φx (J T ) := {φx (J + G) : J ∈ J T }. If x ∈ M \ T then φx (J T ) = φx (G), hence φx (B + G) = φx (Bx + G). On the other hand, if x ∈ T , then φx (J T ) equals φx (J x ), hence we find a P x -compact K such that φx (B − Bx + {Enx (Lxn KLxn )} + G) = 0. Consequently, W x (B−Bx +{Enx (Lxn KLxn )}) = 0, and since W x (B−Bx ) = 0, it even holds K = 0. Therefore, we get again that φx (B + G) = φx (Bx + G). Thus, the local equivalence of B and Bx is proved for every x ∈ M , B ∈ LT follows, and yields the inverse closedness of LT /J T in F T /J T . To show the respective assertions for LT and LT /G simply consult Theorem 2.17. Now, let A ∈ T T be J T -Fredholm and B ∈ LT be a regularizer. Let us show that B ∈ T T . If T = I then T T = F T and there is nothing to prove. Otherwise A =J T TAT+λ(I−T) with λ 6= 0 and we can multiply the equations AB =J T I and BA =J T I with (I − T) from the left (right, resp.) and we get λ(I − T)B =J T λB(I − T) =J T (I − T). Hence, B =J T TB + λ−1 (I − T) and B =J T BT + λ−1 (I − T). Putting the first of these equations into the second one we finally obtain B =J T TBT + λ−1 (I − T) that is B ∈ T T . Therefore the assertions on the inverse closedness of T T , T T /G and T T /J T easily follow as well. Theorems 2.6, 2.17 and Corollary 2.23 show that all snapshots of a (J T -Fredholm / Fredholm / stable) sequence A ∈ T T are (P t -Fredholm / Fredholm / invertible). For the converse let A ∈ T T be such that all snapshots are P t -Fredholm. Due to the local principle of Allan and Douglas in Theorem 3.3 it suffices to check that Φx (A + J T ) is invertible having an inverse Φx (Cx + J T ) with Cx + J T ∈ F T /J T for every x ∈ MC/G , respectively. For x ∈ M we have that Φx (A + J T ) equals Φx (Ax + J T ), and since W x (A) is P x -Fredholm each P x -regularizer B belongs to L(Ex , P x ) by Theorem 1.14. Its lifting Bx := {Enx (Snx BSnx )} belongs to F T and Bx + G is in the commutant of C/G as the above considerations show. Moreover, I − Bx Ax + J T = {Ln − Enx (Snx BSnx )Enx (Snx ASnx )} + J T

= {Ln − Enx (Snx BA(Snx )3 )} + J T = {Ln − Enx ((Snx )4 )} + J T ,

hence Φx (I − Bx Ax + J T ) (as well as Φx (I − Ax Bx + J T )) equals zero. This gives the local invertibility in x ∈ M . Now let x0 ∈ M belong to the closure of MC/G \ M . From Theorem 3.3 we can derive that there is a point y0 ∈ MC/G \M in a neighborhood of x0 such that Φy0 (A+J T ), which equals Φy0 (λI + J T ), is invertible as well, and consequently λ 6= 0. Since λ is independent of y ∈ MC/G \M we get the invertibility of all cosets Φy (A+J T ) and hence the J T -Fredholmness of A. Now, let all snapshots of A ∈ T T be Fredholm (or even invertible). Then each of them is P t Fredholm, respectively, and with the above we can conclude that A is regularly J T -Fredholm. Corollary 2.24 gives the Fredholm property of A, and in case of invertible snapshots we have splitting numbers α(A) = β(A) = 0, hence stability from Theorem 2.21.

68


Remark 3.10. The advantages of the described setting are the following: Suppose that one is interested in a certain subalgebra A ⊂ F T and one wants to eliminate the unwieldy condition on the J T -Fredholm property in the results on stability, convergence of approximation numbers or the index formula for the sequences in A. Then Proposition 3.8 shows that one only needs to find a suitable C and to prove that the generators of A are localizable w.r.t. C, to guarantee that A ⊂ LT . In many applications this simplifies matters essentially because the algebras under consideration are often spanned by only a few rather basic elements. We again refer to the application in Section 4.3 for a telling example. Furthermore, Theorem 3.9 obviates the need for checking the J T -Fredholm property and, moreover, the stated inverse closedness provides control over the respective regularizers and inverses. The latter is a crucial aid in the next section where we deal with the convergence of condition numbers.

3.2.2

Convergence of norms and condition numbers

Within this section suppose that C defines a localizing setting in F T such that, additionally, • kφx (A + G)k ≤ kW x (A)k ≤ kAg + Gg k holds for every A ∈ T T , g ∈ H+ and x ∈ M ,6 • T T /G is a KMS-algebra with respect to C/G. and say that C defines a faithful localizing setting. Proposition 3.11. Let A = {An } ∈ T T . The norms kAn k and kA−1 n k converge and lim kAn k = max kW t (A)k,

n→∞

t∈T

t −1 7 lim kA−1 k. n k = max k(W (A))

n→∞

t∈T

Proof. At first, notice that f : MC/G → R+ , x 7→ kφx (A + G)k is upper semi-continuous on MC/G by Theorem 3.3 and constant on MC/G \ M since A ∈ T T . If MC/G \ M is not empty then choose x0 ∈ M which also belongs to the closure of MC/G \ M and conclude that kφx (A + G)k ≤ kφx0 (A + G)k for all x ∈ MC/G \ M. With the formula (3.4) from Theorem 3.5 this yields an x1 ∈ M such that for every g ∈ H+ sup kW t (A)k ≤ kAg + Gg k = lim sup kAgn k ≤ lim sup kAn k = kA + Gk n→∞

t∈T

n→∞

= max kφx (A + G)k = max kφx (A + G)k = kφx1 (A + G)k x∈MC/G

≤ kW

x1

x∈M

(A)k.

Since there is a t1 ∈ T with W t1 (A) = W x1 (A) we get that limn kAn k exists and is determined as the maximum of the norms of all snapshots. Now we consider the “inverses”. Suppose that one snapshot of A is not invertible. Then, by Theorem 2.17, the sequence A and each of its subsequences cannot be stable, that is there is −1 no bounded subsequence of {A−1 n }, hence limn kAn k = ∞. If, conversely, all snapshots are invertible then Theorem 3.9 yields that A is stable, a sequence whose nth entry is A−1 n (if An is invertible) is a G-regularizer for A in T T and its snapshots are (W t (A))−1 . Apply the above to this sequence to prove the second asserted equation. 6 For 7 We

the definition of the local homomorphisms φx on Com(C/G) see Section 3.1. again use the convention kB −1 k = ∞ if B is not invertible.


69

Corollary 3.12. Let A = {An } ∈ T T be stable. Then the condition numbers of the operators An which are defined by cond(An ) := kAn kkA−1 n k converge and their limit is lim cond(An ) = max kW t (A)k · max k(W t (A))−1 k.

n→∞

t∈T

t∈T

Remark 3.13. Let us give some more concrete criteria for a faithful localizing setting. 1. The first of the stated conditions in its definition is true if, for every t ∈ T , BP t = lim kEnt k = lim kEn−t k = lim kSnt k = 1. n→∞

n→∞

n→∞

2. For the second condition notice that T T /G belongs to the commutant of C/G in the unital Banach algebra F/G. In general C is not completely contained in T T ,8 but the flexibilized Definition 3.6 of the KMS property permits to write the condition in this form. It is particularly fulfilled if, for every {An }, {Bn } ∈ F T and all ϕ = {ϕn }, ψ = {ψn } ∈ C with kϕ + Gk = kψ + Gk = 1 and ϕψ ∈ G, the following holds 9 kϕn An ϕn + ψn Bn ψn k ≤ max{kAn k, kBn k} for every

n ∈ N.

The first part is straightforwardly proved. Indeed, for A ∈ T T and x ∈ M kφx (A + G)k = kφx (Ax + G)k ≤ kAx + Gk = lim sup kEnx (Snx W x (A)Snx )k ≤ kW x (A)k n→∞

and Theorem 1.19 further yields for every g ∈ H+ kW x (A)k ≤ lim inf kEg−x (Agn )Sgxn k ≤ lim sup kEg−x kkAgn kkSgxn k ≤ lim sup kAgn k. n n n→∞

n→∞

n→∞

For the second part notice that for every > 0 there is an integer N such that, for every k ≥ N , kϕk Ak ϕk + ψk Bk ψk k ≤ max{lim sup kAn k, lim sup kBn k} + . n→∞

n→∞

Hence, for every A, B ∈ F T and ϕ, ψ ∈ C with kϕ + Gk = kψ + Gk = 1 and ϕψ ∈ G, kϕAϕ + ψBψ + Gk ≤ max{kA + Gk, kB + Gk}. Now let C ∈ T T and f, g ∈ C be such that the closures of the supports of f + G and g + G are disjoint. Choose ϕ, ψ ∈ C such that the functions ϕ + G, ψ + G are of norm one, have disjoint support and f − ϕf, g − ψg ∈ G. Then k(f + g)C + Gk = kϕ2 f C + ψ 2 gC + Gk = kϕf Cϕ + ψgCψ + Gk ≤ max{kf C + Gk, kgC + Gk}. Since every pair of continuous functions on a compact set having disjoint support can be approximated by pairs of continuous functions having disjoint closed supports, we obtain the KMS property of T T /G with respect to C/G. 8 Fix x ∈ int(M \ T ) and a function ϕ being equal to 1 in x and identically zero on T . Then ϕ + G and ϕx + G are not locally equivalent in x. 9 This idea already appeared in [10].

70


3.2.3

Convergence of pseudospectra

We turn our attention again to the asymptotic behavior of the pseudospectra spN, An which have been introduced and studied in Section 1.4 and we now suppose an additional condition which we adopt from [27] and [84]. Definition 3.14. A Banach space X is called complex uniformly convex if for every > 0 there exists a δ > 0 such that x, y ∈ X,

kyk ≥ and kx + ζyk ≤ 1 for all ζ ∈ C, |ζ| ≤ 1 always implies kxk ≤ 1 − δ.

As already mentioned in Remark 1.51 this simplifies matters essentially. The precise result, taken from Shargorodsky [84], Theorem 2.6 and Corollary 2.7, reads as follows. Theorem 3.15. Let Ω be a connected open subset of C, let X or X∗ be complex uniformly convex, and let A : Ω → L(X) be an analytic operator-valued function. Suppose that there exists a λ0 ∈ Ω such that the derivative A0 (λ0 ) of A in λ0 is invertible. If kA(λ)k ≤ M for all λ ∈ Ω then kA(λ)k < M for all λ ∈ Ω. We note that every Hilbert space is complex uniformly convex (see [17]). For a further class of Banach spaces which are complex uniformly convex (and which are highly important for the present text), we borrow the following again from [84], preliminary to Theorem 2.5. Proposition 3.16. . 1. Let (S, Σ, µ) be an arbitrary measure space and let X = Lp (S, Σ, µ) with 1 ≤ p < ∞ denote the set of all (equivalence classes of ) Lebesgue measurable functions f on (S, Σ, µ) with the p-th power of kf k being Lebesgue integrable. Then X is complex uniformly convex. If X = L∞ (S, Σ, µ) is the space of all Lebesgue measurable and essentially bounded functions then X ∗ is complex uniformly convex. 2. If, additionally, µ(S) < ∞ then lp (Z, X) (or (lp (Z, X))∗ , resp.) is complex uniformly convex. This particularly includes the cases lp (Z, CN ), N ∈ N, where CN is provided with the p-vector norm and the counting measure. Proof. For the proof of X (or X ∗ ) being complex uniformly convex see the mentioned discussion ˆ Σ, ˆ µ in [84]. Let µ(S) < ∞. Then X := lp (Z, X) can be identified with Lp (S, ˆ), where the measure ˆ Σ, ˆ µ space (S, ˆ) is given by ( ) ! [ [ X ˆ := Sˆ := Z × S, Σ {k} × Ak : Ak ∈ Σ and µ ˆ {k} × Ak := µ(Ak ), k∈Z

k∈Z

k∈Z

and the first part applies. Definition 3.17. Let M1 , M2 , . . . be a sequence of non-empty subsets of C. The uniform (partial) limiting set u-lim Mn n→∞

p-lim Mn n→∞

of this sequence is the set of all z ∈ C that are (partial) limits of a sequence (zn ) with zn ∈ Mn . These limiting sets are closed as shown with [30], Proposition 3.2.


71

Proposition 3.18. Let C define a faithful localizing setting and further let all spaces Et or their duals be complex uniformly convex. For A = {An } ∈ T T and every N ∈ Z+ , > 0 [ u-lim spN, An = p-lim spN, An = spN, W t (A). n→∞

n→∞

t∈T

Chapter 3 in the book [13] contains a concise discussion of the development and application of -pseudospectra as well as the present proposition in this particular case N = 0. We particularly point out again that results similar to Theorem 3.15, which guarantee that the resolvent is not locally constant, have been in that business from the very beginning, see Böttchers pioneering work [6], for example. We further should mention [9] which has presented the convergence results for -pseudospectra of the finite sections of operators on Lp -spaces, 1 < p < ∞, for the first time. The present proof of Proposition 3.18 is an adapted variant of [13], Theorem 3.17. N

Proof. Suppose z ∈ sp W t (A), that is W t (A − zI) and hence W t ((A − zI)2 ) are not invertible. N Then A − zI is not stable and Proposition 3.11 yields that limn k(An − zLn )−2 k = ∞ which implies z ∈ spN, An for sufficiently large n, and therefore z ∈ u-limn spN, An . N N So, now suppose that A − zI is stable, but z ∈ spN, W t (A), i.e. k(W t (A) − zI t )−2 k ≥ −2 t t for one t ∈ T . Let U be an arbitrary open ball around z such that W (A) − yI is invertible N N for all y ∈ U . If k(W t (A) − yI t )−2 k would be less than or equal to −2 for every y ∈ U then N N Theorem 3.15 would imply that k(W t (A) − zI t )−2 k < −2 , a contradiction. For this notice N that the function y 7→ (W t (A) − yI t )−2 is analytic on U and its derivative in z is invertible. N N Hence there is a y ∈ U such that k(W t (A) − yI t )−2 k > −2 , that is we can find a k0 such that t −2N

t

k(W (A) − yI )

−2N 1 k> − k

for all k ≥ k0 .

Because U was arbitrary we can choose a sequence (zm )m s.t., for m ≥ 1/, zm belongs to the N (N, − 1/m)-pseudospectrum of W t (A) and zm → z as m → ∞. Since limn k(An − zm Ln )−2 k N exists and equals maxt∈T k(W t (A) − zm I t )−2 k, due to Proposition 3.11, it is greater than or N N N equal to ( − 1/m)−2 . Consequently, for sufficiently large n, k(An − zm I)−2 k ≥ −2 and thus zm ∈ spN, An . This shows that z = limm zm belongs to the closed set u-limn spN, An . N N Finally consider the case that k(W t (A) − zI t )−2 k < −2 for all t ∈ T . Then N

N

N

lim k(An − zLn )−2 k = max k(W t (A) − zI t )−2 k < −2 ,

n→∞

t∈T

N

N

hence there are a δ > 0 and an n0 ∈ N such that k(An − zLn )−2 k ≤ −2 − δ for all n ≥ n0 . If |y − z| is sufficiently small and n ≥ n0 we then further get N

N

k(An − yLn )−2 k = k((An − zLn )(Ln + (z − y)(An − zLn )−1 ))−2 k N

N

= k(An − zLn )−2 (Ln + (z − y)(An − zLn )−1 )−2 k N

≤

N

−2 − δ k(An − zLn )−2 k ≤ N (1 − |z − y|k(An − zLn )−1 k)2 (1 − |z − y|k(An − zLn )−1 k)2N N

< −2 .

Thus, z does not belong to p-limn spN, An . Since u-limn spN, An ⊂ p-limn spN, An this completes the proof.

72


Proposition 3.6 in [30] states that for compact sets Mn the limits u-limn Mn and p-limn Mn coincide if and only if Mn converge w.r.t. the Hausdorff distance (to the same limiting set).10 Thus, we can reformulate the preceding proposition as follows. Corollary 3.19. Let C define a faithful localizing setting and further let all spaces Et or their duals be complex uniformly convex. For a sequence A = {An } ∈ T T the (N, )-pseudospectra of the elements An converge w.r.t. the Hausdorff distance to the union of the (N, )-pseudospectra of all snapshots W t (A). This now clarifies the observation that we made in the beginning of Section 1.4.4: In general, one cannot expect that for a given increasing sequence (Tn ) of projections the (N, )-pseudospectra of the truncations Tn ATn of an operator A converge to spN, A. The reason is that A captures only one facet of the asymptotic behavior of (Tn ATn ), and (at least in many cases) further snapshots are needed to restore all relevant details of this sequence. We finally pose the question what happens if the spaces Et are not known to be complex uniformly convex, and Theorem 3.15 is not available. A thorough look into the proof of Proposition 3.18 reveals that this additional condition is only needed to conclude z ∈ u-limn spN, An for suffiN N ciently large n from k(W t (A) − zI t )−2 k ≥ −2 for one t ∈ T . Without it, we can arrive at the N N same conclusion if we suppose that k(W t (A) − zI t )−2 k is strictly greater than −2 for one t. T Therefore, the picture for general Banach spaces is as follows: For every A ∈ T , every N ∈ Z+ and 0 < δ < [ [ spN,δ W t (A) ⊂ u-lim spN, An ⊂ p-lim spN, An ⊂ spN, W t (A). t∈T

n→∞

n→∞

t∈T

Thus we can use Theorem 1.44 to carry forward Theorem 1.48: Corollary 3.20. Suppose that the index set T of the algebraic framework F T is finite and C defines a faithful localizing setting. Let A = {An } ∈ T T . For every 0 < α < < β there is an N0 ∈ Z+ such that, for every N ≥ N0 , ! ! [ [ t t Bα sp W (A) ⊂ u-lim spN, An ⊂ p-lim spN, An ⊂ Bβ sp W (A) . t∈T

n→∞

n→∞

t∈T

Remark 3.21. Such results on the convergence of norms, condition numbers and pseudospectra have been proved for various classes of operators, such as Wiener-Hopf operators [4], Toeplitz operators [13], [8], band-dominated operators on l2 [57], or convolution type operators on cones [50]. A systematic and abstract presentation for the C ∗ -algebra case is in [30]. The present text provides several extensions: It is in a more abstract and versatile axiomatic formulation, which covers more exotic situations such as l∞ spaces, in particular the class of band-dominated operators on all lp -spaces as we will see in Section 3.2.5. Moreover, it treats (N, )-pseudospectra, and it overcomes one of the most unpleasant hurdles in that business: the inverse closedness of the algebras under consideration is automatically given.

3.2.4

Rich sequences meet localization

Let us condense the previous results and apply them to rich sequences A. For this, HA again stands for the set of all strictly increasing sequences g of positive integers such that Ag is a T -structured subsequence of A. Further, we denote by T RT the set of all sequences A ∈ RT 10 This

observation is due to Hausdorff and can be found in his book [35] as well.


73

which have the following property: Every T -structured subsequence Ag has a subsequence Ah and an algebra C (that may depend on A and h) which defines a faithful localizing setting (in FhT ), such that Ah ∈ ThT . Theorem 3.22. A sequence A ∈ T RT is Fredholm, if and only if all its snapshots are Fredholm. In this case ! ! X X t t α(A) = max dim ker W (Ag ) , β(A) = max dim coker W (Ag ) . g∈HA

g∈HA

t∈T

t∈T

Furthermore, for every A = {An } ∈ T RT , lim sup kAn k = max max kW t (Ag )k,

t −1 lim sup kA−1 k, n k = max max k(W (Ag ))

g∈HA t∈T

n→∞

lim inf kAn k = min max kW (Ag )k, n→∞

g∈HA t∈T

n→∞

t

lim inf n→∞

g∈HA t∈T

kA−1 n k

= min max k(W t (Ag ))−1 k, g∈HA t∈T

and if, additionally, all spaces Et or their duals are complex uniformly convex then [ p-lim spN, An = spN, W t (Ag ). n→∞

g∈HA , t∈T

A sequence A ∈ T RT is stable if and only if all its snapshots are invertible. In this case t t −1 lim sup cond(An ) = max max kW (Ag )k · max k(W (Ag )) k , g∈HA t∈T t∈T n→∞ t t −1 lim inf cond(An ) = min max kW (Ag )k · max k(W (Ag )) k . n→∞

g∈HA

t∈T

t∈T

Proof. Let all snapshots of A ∈ T RT be Fredholm. Every T -structured subsequence Ag of A has a subsequence Ah which belongs to ThT , thus Theorem 3.9 yields that Ah is Fredholm and regularly JhT -Fredholm. Now, we derive the Fredholm property of A and formulas for α(A) and β(A) from Theorem 2.26. For the converse Corollary 2.23 can be employed. The stability can be tackled in the same way. Recall from Proposition 3.11 that, for every g ∈ HA and the respective localizable subsequence Ah , max kW t (Ag )k = max kW t (Ah )k = lim kAhn k ≤ lim sup kAn k. t∈T

t∈T

n→∞

n→∞

Now, choose a sequence j ∈ H+ such that limn kAjn k = lim supn kAn k and pass to a subsequence g ∈ HA of j to see maxt∈T kW t (Ag )k = limn kAjn k = lim supn kAn k. For the other equations we proceed analogously, with the aid of Propositions 3.11, 3.18 and Corollary 3.12.

3.2.5

Example: finite sections of band-dominated operators

Back in the lp setting, we want to apply the latest tools to the sequences in the algebra FAlp which is generated by the finite section sequences of all rich band-dominated operators. By doing this, we recover and extend the observations of [63], Section 6.3,11 as well as those of Section 2.5 in the present text. 11 See

also [69] and the preprint [57].

74


bγ1

1 −1

1 ϕ

bγn

1 −n

n − γn n

ϕ ◦ bγn

Figure 3.1: Inflating a function ϕ ∈ C[−1, 1] via the splines bγn . Theorem 3.23. Let A = {An } ∈ FAlp and g ∈ HA . Then Ag is J T -Fredholm if and only if its snapshots W t (Ag ) are P t -Fredholm, respectively. We further have the limits C1 := lim kAgn k = max{kW (A)k, kW 1 (Ag )k, kW −1 (Ag )k}, n→∞

−1 C2 := lim kA−1 k, k(W 1 (Ag ))−1 k, k(W −1 (Ag ))−1 k}, gn k = max{k(W (A)) n→∞

hence limn cond(Agn ) = C1 C2 in the case of a stable sequence. If, additionally, lp or (lp )∗ is complex uniformly convex then the (N, )-pseudospectra of the operators Agn converge w.r.t. the Hausdorff distance to the union of the (N, )-pseudospectra of the snapshots.12 Proof. The outline of this proof is very simple and straightforward: We are going to introduce an algebra C γ which defines a faithful globally localizing setting in FgT and turns Ag into a C γ localizable sequence, hence allows to apply the results on the convergence of norms, condition numbers and pseudospectra. First of all, we set i1 := 1 and choose a subsequence i = (ik ) of g such that n o 1 t (t) t sup k(A(t) − W (A ))L k, kL (A − W (A ))k : t ∈ T, g ≥ i < g k k g n k gn gn k

(3.6)

(t)

for every k ≥ 2. This is possible since Agn tends P t -strongly to W t (Ag ). Now, for k ∈ N, define γn := k/2 for all n ∈ {ik , . . . , ik+1 − 1}, and let bγn : R → [−1, 1] denote the continuous piecewise linear splines given by bγn (±n) = ±1 and bγn (±(n − γn )) = ±1/2 which are constant outside the interval [−n, n], respectively. For every continuous function ϕ ∈ C[−1, 1] we let ϕγn stand for the restriction of the function ϕ ◦ bγn to Z. Thus, the sequence {ϕγn Ln } consists of operators of multiplication by inflated copies of ϕ with a certain distortion (see Figure 3.1). With k{ϕγn Ln }k ≤ kϕk∞ it straightforwardly follows that the set C γ := {{ϕγn Ln } : ϕ ∈ C[−1, 1]} forms a commutative subalgebra of F T , where W t ({ϕγn Ln }) = ϕ(t)I t for every t and every ϕ. By ∗ : {ϕγn Ln } 7→ {ϕγn Ln } an involution is given and the mapping ϕ 7→ {ϕγn Ln } + G is a ∗-homomorphism from C[−1, 1] onto C γ /G. We show that this mapping is even isometric, and therefore C γ /G proves to be a unital commutative C ∗ -algebra and its maximal ideal space can be identified with the interval [−1, 1], due to Theorem 3.2. For this, we note that kϕk∞ ≥ k{ϕγn Ln }kF ≥ k{ϕγn Ln } + GkF /G holds and, on the other hand, kϕk∞ = sup{|ϕ(x)| : x ∈ [−1, 1]} ≤ k{ϕγn Ln } + GkF /G follows 12 Without

the complex uniform convexity Corollary 3.20 is still applicable.


from

75

13

|ϕ(x)| = kϕ(x)Ik = kP-limV−b(bγn )−1 (x)c ϕγn Ln Vb(bγn )−1 (x)c k n→∞

≤ lim inf kV−b(bγn )−1 (x)c ϕγn Ln Vb(bγn )−1 (x)c k ≤ lim sup kϕγn Ln k = k{ϕγn Ln } + GkF /G . n→∞

n→∞

Set M := [−1, 1]. To every x ∈ (−1, 1) we associate the direction t(x) := 0 ∈ T , and for every x ∈ [−1, 1] we fix a function Fx ∈ C[−1, 1] which equals 1 in a neighborhood of x, equals 0 in a neighborhood of every t ∈ T \ {x} and takes only real values between 0 and 1. Then C γ defines a faithful globally localizing setting. For this apply Remark 3.13 and notice that k(ϕγn An ϕγn + ψnγ Bn ψnγ )ak = max{kϕγn An ϕγn ak, kψnγ Bn ψnγ ak} ≤ max{kAn k, kBn k}kak

(3.7)

holds for all a ∈ En , all {An }, {Bn } ∈ F and all ϕ, ψ ∈ C[−1, 1] of norm 1 with disjoint support in the case p = ∞. For 1 ≤ p < ∞ we replace (3.7) by k(ϕγn An ϕγn + ψnγ Bn ψnγ )akp = kϕγn An ϕγn akp + kψnγ Bn ψnγ akp

≤ kϕγn An kp kϕγn akp + kψnγ Bn kp kψnγ akp ≤ max{kAn kp , kBn kp }kakp .

It remains to show that Ag ∈ TgT = LTg . For this assume that we can show, for every B ∈ Alp , that lim sup k[V−k BVk , ϕγn ]k = 0 for every {ϕγn Ln } ∈ C γ . (3.8) n→∞ k∈Z

Then FAlp /G is obviously contained in the closed subalgebra Com(C γ /G) of F/G which yields that Ag + Gg commutes with every element of Cgγ /Gg . Every snapshot W x (Ag ) is again banddominated (or at least the compression of a band-dominated operator to the space Ex ) and, due to the choice of the functions Fxg it is always liftable. Furthermore, by (3.8), it almost commutes with Fxg and Axg + Gg belongs to Com(Cgγ /Gg ) as well. Moreover, for every x ∈ (−1, 1), equation (3.8) provides that sequences of the form ((I − Ln )BFxn ) and (Fxn B(I − Ln )), with B being band-dominated, tend to zero in the norm as n → ∞, hence Ag + Gg and Axg + Gg easily prove to be locally equivalent in x. In case x = 1 (or in an analogous way also for x = −1) we choose ϕ ∈ C[−1, 1] equal to 1 in a neighborhood of x and identically zero on [−1, 1/2] (or [−1/2, 1], respectively). Then {ϕγgn Lgn }(Ag − Axg ) ∈ Gg by the construction of the (γn ) and (3.6), which yields the local equivalence in x again. Thus Ag ∈ LTg , and Theorem 3.9, Proposition 3.11 and Corollaries 3.12, 3.19 apply and give the assertions of the theorem. So, let us check the relation (3.8) for band-dominated operators B. It is easily proved for every shift operator and every operator of multiplication, hence it is clear for band operators and follows for band-dominated ones by a simple approximation argument. Clearly, for every sequence A ∈ FAlp , the proof of the previous theorem shows that A ∈ T RT , hence Theorem 3.22 applies, recovers and completes Theorem 2.32. More precisely, for every A = {An } ∈ FAlp , we have formulas for the lim supn and the lim inf n of both kAn k and kA−1 n k, in terms of norms of the snapshots and their inverses. In case of a stable sequence, this is true for cond(An ) as well. Moreover, if the space lp or its dual is complex uniformly convex then, also by Theorem 3.22, [ p-lim spN, An = spN, W t (Ag ).14 n→∞

(bγn )−1 being the inverse of the bijective function bγn : [−n, n] → [−1, 1], and b·c the floor function. a related observation in case p = 2 see [63], Theorem 6.3.15.

13 With 14 For

g∈HA , t∈T

76


3.3

P-algebras

This section is devoted to the theory of Banach algebras with approximate projections. In particular, we give the pending proofs for the results that we stated in Section 1.2.1.

3.3.1

Basic definitions

Definition 3.24. Let A be a unital Banach algebra and let P = (Pn )n∈N ⊂ A be a bounded sequence of elements with the following properties: • Pn 6= I for all n ∈ N, and Pn 6= 0 for large n,

• For every m ∈ N there is an Nm ∈ N such that Pn Pm = Pm Pn = Pm if n ≥ Nm .

Then P is referred to as an approximate projection. In all what follows we set Qn := I − Pn and we further write m n if Pk Ql = Ql Pk = 0 for all k ≤ m and all l ≥ n. P-compactness Let P be an approximate projection on A. An element K ∈ A is called Pcompact if kKPn − Kk and kPn K − Kk tend to zero as n → ∞. By A(P, K) we denote the set of all P-compact elements and by A(P) the set of all elements A ∈ A for which AK and KA are P-compact whenever K is P-compact.

Theorem 3.25. Let P be an approximate projection. A(P) is a closed subalgebra of A, it contains the identity, and A(P, K) is a proper closed ideal of A(P). An element A ∈ A belongs to A(P) if and only if, for every k ∈ N, kPk AQn k → 0

and

kQn APk k → 0

as

n → ∞.

(3.9)

Proof. The following proof is an adaption of [63], Proposition 1.1.8. The conditions (3.9) are clearly necessary for A ∈ A(P). Let, conversely, A satisfy (3.9) and let K ∈ A(P, K). Given > 0, choose k such that kQk Kk < , and choose N such that kQn APk k < for all n ≥ N . Then kQn AKk ≤ kQn AkkQk Kk + kQn APk kkKk < (kQn Ak + kKk) for all n ≥ N . The other conditions can be checked similarly, thus A ∈ A(P). It is immediate from the definition that A(P, K), A(P) are subalgebras of A and that A(P, K) forms a proper ideal in A(P). To check the closedness of A(P, K) let (Kn ) ⊂ A(P, K) converge to K ∈ A in the norm, and consider kQn Kk ≤ kQn Km k + kQn kkK − Km k. The latter term becomes as small as desired, if m is large, and the first one tends to zero as n → ∞ for every m. Analogously, we find that kKQn k → 0 as n → ∞, hence K ∈ A(P, K). If A is the (norm) limit of (An ) ⊂ A(P) then AK, KA are the limits of (An K), (KAn ) ⊂ A(P, K) for every K ∈ A(P, K), hence are P-compact as well. Thus, A ∈ A(P). P-Fredholmness Here are two possible ways to introduce “almost invertibility”, in analogy to the Fredholm property in operator algebras. • For A ∈ A we may assume that there is a P-regularizer B ∈ A, that is I − AB, I − BA are P-compact. • For A ∈ A(P) we may assume that the coset A + A(P, K) is invertible in the quotient algebra A(P)/A(P, K).

The big question is, if these definitions coincide for A ∈ A(P).

3.3. P-ALGEBRAS

3.3.2

77

Uniform approximate projections and the P-Fredholm property

Definition 3.26. Given an algebra A with an approximate projection P = (Pn ), we set S1 := P1 and Sn P:= Pn − Pn−1 for n > 1. Furthermore, for every finite subset U ⊂ N, we define PU := k∈U Sk . P is called uniform if CP := sup kPU k < ∞, the supremum over all finite U ⊂ N. Two approximate projections P = (Pn ) and Pˆ = (Fn ) on A are said to be equivalent if for every m ∈ N there is an n ∈ N such that Pm Fn = Fn Pm = Pm

and Fm Pn = Pn Fm = Fm .

ˆ one easily checks that A(P, K) = A(P, ˆ K), In case of equivalent approximate projections P and P, ˆ hence also A(P) = A(P). Theorem 3.27. Let P be a uniform approximate projection on A and A ∈ A(P). Then there is an equivalent uniform approximate projection Pˆ = (Fn ) on A with CPˆ ≤ CP such that k[Fn , A]k = kAFn − Fn Ak → 0

as

n → ∞.

Proof. Successively choose integers 1 = j1 i2 j2 i3 j3 . . . such that for every l kPs AQjl k ≤ (2l+2 l)−1 ∀ s ≤ il

and kQt APjl k ≤ (2l+5 l)−1 ∀ t ≥ il+1 .

(3.10)

Then, for all k, n ∈ N set Ukn := {i2n +k−1 + 1, . . . , i2n +k } and Vkn := {j2n +k−2 + 1, . . . , j2n +k }, as well as U0n := {1, . . . , i2n } and V0n := {1, . . . , j2n }, and find kPUkn APj2n +k−2 k ≤ (2k+2 n)−1

and kPUkn AQj2n +k k ≤ (2k+2 n)−1 .

Thus kPUkn AQVkn k ≤ (2k+1 n)−1 , yielding For n ∈ N we set Fn :=

n−1 X k=0

1−

k n

X

k∈Z+

kPUkn AQVkn k ≤

1 . n

(3.11)

PUkn

and deduce that Fn Fn+1 = Fn+1 Fn = Fn as well as

n

n

n X n

X k

X

1 1 X

≤ CP , n n n PUn−k kPUn−k PUn−k kFn k =

=

≤

n n n j=1

k=1

k=1

(3.12)

k=j

that is Pˆ = (Fn ) is an approximate projection. Further, P and Pˆ are equivalent, since Pj2n −1 Fn = Fn Pj2n −1 = Pj2n −1

and Fn Pj2n +n−1 = Pj2n +n−1 Fn = Fn .

For bounded subsets U ⊂ R we finally introduce the elements FU as in Definition 3.26 and easily check that they can be represented in the form FU =

N −1 X k=0

k PWk N

78


with certain disjoint finite sets Wk ⊂ N, k = 0, . . . , N − 1. By an estimate similar to (3.12) we deduce that Pˆ is uniform with CPˆ ≤ CP . It remains to consider the commutators of A and Fn . As a start, notice that AFn = =

n X

k=0 n X

PUkn AFn + Qi2n +n AFn PUkn APVkn Fn +

n X

PUkn AQVkn Fn + Qi2n +n APj2n +n−1 Fn ,

k=0

k=0

where the last term is less than CP n−1 in the norm and the middle term is not greater than CP n−1 as well, due to the above estimate (3.11). The first one equals X n k k I + PUkn APVkn P AP Fn − 1 − 1− n n k=0 k=0 n n X X k 1 n n − PUk+1 = PUkn APVkn (PUk−1 )− 1− PUkn AQVkn + Fn A, n n n X

Ukn

Vkn

k=0

k=0

n = PU n where we redefine PU−1 := 0. The second item is smaller than n−1 in the norm, thus n+1

n

2C + 1 1

X n

P n n PUk A(I − QVkn )(PUk−1 − PUk+1 ) + kAFn − Fn Ak ≤

n n

k=0

n

2C n−1 + 2C + 1 1

X n

P P n n ≤ PUk A(PUk−1 . − PUk+1 ) +

n n k=0

Now we obviously have

n 2

X

X

n PUkn APUk−1

≤

k=0 l=0

k≡l

n X

k=0 mod 3

n PUkn APUk−1

and this can be further estimated by

     

2 n n n n X  X X X  X       

 

 n n n n P A P + P A − P Uk Uj−1  Uj−1  

Uk   

j=0 l=0 k=0 k=0 j=0,j6 = k

k≡l(3)

k≡l(3)

j≡l(3) j≡l(3)



  

n

n

2 n n X  X X

 X 

X

 

 

 n n n n n ≤ P A P + P AQ P Uk  Uj−1  Uk Vk Uj−1  .



j=0,j6=k

j=0 l=0 k=0 k=0

k≡l(3)

k≡l(3)

j≡l(3)

j≡l(3) We conclude

n 2

X

X

n PUkn APUk−1 CP kAkCP + n−1 CP = 3CP (kAkCP + n−1 ),

≤

k=0

l=0

3.3. P-ALGEBRAS

79

and with a similar estimate for

P

kAFn − Fn Ak ≤

n we finally get the assertion by PUkn APUk+1

6CP (kAkCP + n−1 ) + 2CP n−1 + 2CP + 1 . n

Now, we can answer the question on the correct definition of P-Fredholmness affirmatively. Theorem 3.28. Let P = (Pn ) be a uniform approximate projection on A and A ∈ A(P). Then the coset A + A(P, K) is invertible in the quotient algebra A(P)/A(P, K) if and only if there is a P-regularizer B ∈ A for A. In particular, every P-regularizer for A ∈ A(P) (if it exists) belongs to A(P), and A(P) is inverse closed in A. Proof. Obviously, the invertibility of the coset implies the existence of a P-regularizer B. Conversely, let B ∈ A be such that P := I − BA, P 0 := I − AB ∈ A(P, K). Theorem 3.27 yields an equivalent uniform approximate projection Pˆ = (Fn ) such that k[A, Fn ]k → 0 as n → ∞. Thus, for Gn := I − Fn , we also have k[A, Gn ]k → 0 as n → ∞. Notice that kP Gn k and kGn P 0 k tend to zero as n → ∞, hence (writing An ∼n Bn if kAn − Bn k → 0 as n → ∞) BGn ∼n BGn AB ∼n BAGn B ∼n Gn B. Therefore k[B, Gn ]k → 0 as n → ∞, and we conclude, for every k, that kFk BGn k and kGn BFk k ˆ = A(P) due to Theorem 3.25. tend to zero as n → ∞. Thus B ∈ A(P) These results are completely new and clear up the open problem which has been in the background of the limit operator method [63], [44] for several years. They were recently published in [81] for the case of operator algebras L(X, P). This consistent picture can now be condensed into the following definition. Definition 3.29. We say that A ∈ A(P) is P-Fredholm if there is a P-regularizer B ∈ A for A.

3.3.3

P-centralized elements

We shortly discuss a class of elements which may be regarded as analogues to band-dominated operators in operator algebras, in a sense. Definition 3.30. Let P = (Pn ) be a uniform approximate projection on the Banach algebra A such that Pn+1 Pn = Pn Pn+1 = Pn for every n ∈ N. Say that A ∈ A is P-centralized if lim sup{kPk AQk+m k, kQk+m APk k : k ∈ N} = 0.

m→∞

Proposition 3.31. The set A(P, C) of all P-centralized elements is a closed unital subalgebra of A(P) and contains A(P, K) as a closed ideal. If A ∈ A(P, C) is P-Fredholm then each of its P-regularizers is P-centralized as well. In particular, A(P, C) is inverse closed in A and A(P). This observation will be picked up in the applications part 4.4.6 in order to extend the results on the Fredholm property and the stability of the finite sections of band-dominated operators to a larger class of operators which will be called weakly band-dominated operators.

80


Proof. Apply Theorem 3.25 to get A(P, C) ⊂ A(P), and straightforwardly check that A(P, C) is even a closed linear subspace of A(P) containing A(P, K) and the identity. The estimate kPk ABQk+2m k ≤ kPk AkkPk+m BQk+2m k + kPk AQk+m kkBQk+2m k and its dual for kQk+2m ABPk k obviates that A(P, C) is an algebra. Let B ∈ A be a P-regularizer for A ∈ A(P, C). By Theorem 3.28, B already belongs to A(P), and in fact the same arguments lead to the proof of B belonging to A(P, C): Since A is P-centralized, we can replace (3.10) in the proof of Theorem 3.27 by the stronger conditions sup kPs+r AQjl +r k ≤ (2l+2 l)−1 ∀ s ≤ il

and

r∈Z+

sup kQt+r APjl +r k ≤ (2l+5 l)−1 ∀ t ≥ il+1

r∈Z+

and, instead of constructing only one approximate projection Pˆ = (Fn ), we now consider the family Pˆ r = (Fnr ), r ∈ N, given by n−1 X k r PUkn +r , Fn := Pr + 1− n k=0

where U + r := {u + r : u ∈ U }. Then we again find that these Pˆ r are uniform approximate projections with CPˆ r ≤ CP and supr∈N k[Fnr , A]k → 0 as n → ∞. Moreover, there is a sequence (mn ) of integers mn ≥ n such that Pmn +r Fnr = Fnr Pmn +r = Pmn +r

and Pmn+1 +r Fnr = Fnr Pmn+1 +r = Fnr

for all n, r ∈ N. Now, set Grn := I − Fnr and check that supr∈N k[Grn , B]k → 0 if n → ∞ as it was done in the proof of Theorem 3.28. Thus, we find that for every fixed n and all m ≥ mn+1 sup kPk BQk+m k = sup kPk BGkn Qk+m k ≤ sup kPk Gkn kkBQk+m k + sup kPk kk[Gkn , B]kkQk+m k, k∈N

k∈N

k∈N

k∈N

Pk Gkn

where the first summand vanishes since = 0 and the second one becomes arbitrarily small as n becomes large. With a dual estimate for supk kQk+m BPk k this provides B ∈ A(P, C) and easily finishes the proof.

3.3.4

P-strong convergence

Let P = (Pn ) be an approximate projection in the unital Banach algebra A. A sequence (An ) ⊂ A converges P-strongly to A ∈ A if, for all K ∈ A(P, K), both k(An − A)Kk and kK(An − A)k tend to 0 as n → ∞. In this case we write An → A P-strongly or A = P-limn An . As for Proposition 1.16 we check that a bounded sequence (An ) in A converges P-strongly to A ∈ A if and only if k(An − A)Pm k → 0 and kPm (An − A)k → 0 for every fixed Pm ∈ P. Definition 3.32. By F(A, P) we denote the set of all bounded sequences (An ) ⊂ A, which possess a P-strong limit in A(P). Furthermore, say that P is an approximate identity if there is a universal 15 constant DP > 0 such that kAk ≤ DP supn kAn k for every sequence (An ) ∈ F(A, P) and the respective P-strong limit A. Remark 3.33. Notice that P is an approximate identity on A if and only if there is a universal constant D > 0 such that kAk ≤ D supm kPm Ak for every A ∈ A(P). Indeed, the implication only if is clear with An := Pn A, and the if part follows with (An ) ∈ F(A, P) from kAk ≤ D sup kPm Ak = D sup lim kPm An k ≤ D sup sup kPm An k ≤ D sup kPm k sup kAn k. m∈N

15 That

m∈N n∈N

is, it is independent of the sequence (An ).

m∈N n∈N

m∈N

n∈N

3.3. P-ALGEBRAS

81

Theorem 3.34. Let P be an approximate identity on the Banach algebra A with identity I. • If (An ) ⊂ A(P) converges P-strongly to A ∈ A then (An ) is bounded and A ∈ A(P), i.e. (An ) ∈ F(A, P). • The P-strong limit of every (An ) ∈ F(A, P) is uniquely determined. • Provided with the operations α(An ) + β(Bn ) := (αAn + βBn ), (An )(Bn ) := (An Bn ), and the norm k(An )k := supn kAn k, F(A, P) becomes a Banach algebra with identity I := (I). The mapping F(A, P) → A(P) which sends (An ) to its limit A = P-limn An is a unital algebra homomorphism and kAk ≤ DP lim inf kAn k. (3.13) n→∞

• Every approximate projection Pˆ which is equivalent to P forms an approximate identity ˆ = F(A, P) holds true. and F(A, P) • Let P be uniform. If (An ) ∈ F(A, P) has an invertible limit A and (Bn ) ⊂ A is a bounded sequence such that (An Bn ), (Bn An ) converge P-strongly to the identity, then (Bn ) belongs to F(A, P) with the limit A−1 . Proof. For the first assertion let K ∈ A(P, K). Then we have An K, KAn ∈ A(P, K) and further kAn K − AKk, kKAn − KAk → 0. Since A(P, K) is closed, this implies that AK and KA belong to A(P, K), i.e. A is in A(P). Assume that (An ) is unbounded. Set BP := supn kPn k and m0 := 1, and successively choose mi−1 ki li pi mi for all i ∈ N as follows: Given mi−1 fix ki mi−1 and choose li ki such that kQki Ali k ≥ 2DP BP i3 . This is possible since (An )n is unbounded and (Pm An )n is bounded, hence (Qm An )n is unbounded for every fixed m. Since P is an approximate identity there is an n such that kQki Ali k ≤ 2DP kPn Qki Ali k by Remark 3.33. Now choose pi n, pi li and mi pi such that kQki Ali k ≤ 2DP kPn Ppi Qki Ali k ≤ 2DP BP kPpi Qki Ali k, P∞ −2 hence kPpi Qki Ali k ≥ i3 . The series (Pmn − Pmn−1 ) converges absolutely, hence it n=1 n defines an element P in the Banach algebra A. This element easily proves to be P-compact. Thus, (P An )n converges to P A in the norm. Therefore (P An )n is bounded, which contradicts kP Aln k ≥

n−2 kQkn Ppn Pmn Aln k kQk Pp Al k n kQkn Ppn P Aln k ≥ = 2 n n n ≥ . kQkn Ppn k (BP + 1)BP n (BP + 1)BP (BP + 1)BP

The second assertion is quite obvious: If kPm (An − A)k and kPm (An − B)k tend to zero as n goes to infinity for every fixed m, then Pm (A − B) = 0 for every m, hence A − B = 0 by Remark 3.33. The proof of F(A, P) being a normed algebra is straightforward, and we only note that if (An ), (Bn ) are bounded and converge P-strongly to A, B ∈ A(P), respectively, then kK(An Bn − AB)k ≤ kK(An − A)Bn k + kKA(Bn − B)k → 0 for every K ∈ A(P, K), as n → ∞, since A ∈ A(P) implies KA ∈ A(P, K). The estimate (3.13) follows from the definition, since we can replace (An ) by one of its subsequences which realize the lim inf and then cancel arbitrarily but finitely many elements at the beginning of this subsequence. Finally, let ((Cnm ))m be a Cauchy sequence of sequences (Cnm ) ∈ F(A, P), where C m shall denote the P-strong limit of (Cnm ), respectively. For every n, (Cnm )m converges in A to an element Cn , and the sequence (Cn ) is uniformly bounded. Furthermore, (3.13) yields that (C m )m is a Cauchy

82


sequence with a limit C ∈ A(P). Now one easily checks that (Cn ) converges P-strongly to C, thus F(A, P) is complete. ˆ K), as well as Remark 3.33 with For the fourth assertion apply the fact that A(P, K) = A(P, kAPm k ≤ kAFk kkPm k and kAFm k ≤ kAPk kkFm k for fixed m and sufficiently large k. Given the sequence (Bn ) in the last assertion, we set B := A−1 , find that B ∈ A(P) by Theorem 3.28, and consequently K(Bn − B) = KB(A − An )Bn + KB(An Bn − I) tend to zero in the norm as n → ∞ for every K ∈ A(P, K). The same holds true for (Bn − B)K and thus (Bn ) ∈ F(A, P) with limit B.

3.3.5

A look into the crystal ball

Notice that it would have been sufficient for this thesis to prove these theorems for the algebras L(X, P) of operators in Part 1. Nevertheless, one argument for the present Section 3.3 is given by its simplicity and beauty. Working on this abstract level reveals that these results are purely Banach algebraic. There is no need for additional tools or notions coming from functional analysis or operator theory. Another reason is that there is a natural connection between the local principle described in Section 3.1 and the present P-algebraic framework. This will not be elaborated here in more detail, as it is not important for the classes of operators we have in mind here. We only mention that there are further families of operators and respective discretization methods in the literature - such as finite sections of Toeplitz operators having piecewise continuous symbol (see [29], [74]), or Cauchy singular integral operators and the collocation method (see [37], [38]) - for which the situation is much more involved, since there one needs to consider the algebras “modulo” J T instead of G to achieve a sufficiently large center for the localization. Maybe the following observation can give a new perspective onto these topics and can contribute some better understanding of inverse closedness: Let A be a unital Banach algebra and C be a commutative C ∗ -subalgebra, not necessarily in the center. We again think of the elements ϕ ∈ C as continuous functions on MC . Now, for a fixed x ∈ MC , we suppose that there is a sequence (ϕxn ) ⊂ C of functions of norm one, being equal to 1 in a neighborhood of x and having nested and contracting supports in the sense that ϕxn+1 ϕxn = ϕxn+1 and, for every ψ ∈ C which is equal to one in a neighborhood of x, there is an n such that ϕxn ψ = ϕxn . Then P x := (Lxn ) with Lxn := 1 − ϕxn is a uniform approximate projection on A and an element A ∈ A is P x -compact iff it is locally equivalent to zero in x in the sense that the norms kψAk and kAψk are small for functions ψ with kψk = ψ(x) = 1 and small support. Denote the set of such elements by Jx . A(P x ) is the subset of A for which Jx gives an ideal. Furthermore, A ∈ A(P x ) is P x -Fredholm iff there is a B ∈ A such that I, AB and BA are locally equivalent in x. In this case B automatically belongs to A(P x ), too. Moreover, for given A ∈ A(P x ) we can replace P x by an approximate projection, consisting of functions in C again, which asymptotically commutes with A. Probably, these observations could lead to another abstract local principle with lower requirements on commutativity, in a sense. At the moment this is just a conjecture, but it is to be hoped that some future work will clear it up.

Part 4

Applications Here we reap the fruit of our labor. We particularly want to demonstrate that the algebraic framework of the previous parts unifies several approaches which have been considered in the past, recovers lots of former results for special classes of operators and permits to extend them. Basically, these extensions are made into three directions: 1. Include more exotic cases such as p = 1, ∞ which stayed out of reach in most applications until now, and obliterate differences to the Hilbert space case and the case of reflexive Banach spaces with ∗-strong convergence. 2. Cover larger classes of operators, mainly by the notion of rich sequences. 3. State some new results, particularly on spectral approximation. Firstly, we pick up again the operators of Section 1.4.4, being quasi-diagonal with respect to a certain sequence (Ln ) of projections, and we have a look at their finite sections with respect to (Ln ). Section 4.2 is devoted to band-dominated operators on Lp -spaces and the whole bundle of results which we already know for their discrete analogues on lp . We particularly rediscover the index formula for locally compact operators of Rabinovich, Roch and Roe [59], [58] and extend it into several directions. The multi-dimensional convolution operators, which are subject of Section 4.3, have been intensively studied by Mascarenhas and Silbermann [48], [49], [50]. The present approach now permits to modify their proofs and to obtain the results for larger classes of such operators on a larger scale of spaces. In the final Section 4.4 we briefly address some more concrete subclasses of Alp and, moreover, we want to catch a glimpse of further relatives beyond the world of band-dominated operators.

84

PART 4. APPLICATIONS

4.1

Quasi-diagonal operators

Definition 4.1. Let X be a Banach space. Let further A ∈ L(X) and assume that there is a bounded sequence (Ln ) of projections on X, such that k[A, Ln ]k := kALn − Ln Ak → 0 as n → ∞.

(4.1)

Then A is said to be quasi-diagonal with respect to (Ln ).1 Proposition 4.2. Let A ∈ L(X) be a quasi-diagonal operator w.r.t. (Ln ). 1. If A is invertible and k[A, Ln ]k < (kLn k2 kA−1 k)−1 then Ln ALn is invertible in L(im Ln ). 2. If A is invertible, k[A, Ln ]k < (kLn k2 kA−1 k)−1 , y ∈ X is given, x ∈ X is the unique solution to Ax = y, and xn ∈ im Ln is the unique solution to Ln ALn xn = Ln y, then kx − xn k ≤ kA−1 k (k[A, Ln ]kkxn k + kLn y − yk) . Proof. This proof is a slight modification of the one given with [14], Proposition 4.2. The estimate kLn ALn A−1 Ln − Ln k = kLn (ALn − Ln A)A−1 Ln k ≤ kLn k2 kA−1 kk[A, Ln ]k < 1

(4.2)

yields the invertibility of Ln ALn A−1 Ln by a Neumann argument, hence the invertibility of Ln ALn from the right. With a similar estimate the invertibility from the left can be shown as well. The second assertion is also straightforward with xn = Ln xn and: kxn − xk ≤ kA−1 kkA(xn − x)k = kA−1 kkALn xn − Ln ALn xn + Ln y − yk ≤ kA−1 k (k(ALn − Ln A)Ln xn k + kLn y − yk) .

Corollary 4.3. Let A ∈ L(X) be a quasi-diagonal operator w.r.t. (Ln ) and let y ∈ X. If A is invertible and k(I − Ln )yk → 0 as n → ∞ then (Ln ALn ) is stable and the solutions xn of the equations Ln ALn xn = Ln y converge in the norm to the solution x of Ax = y. Proof. It only remains to mention that for all n such that k[A, Ln ]k < (2kLn k2 kA−1 k)−1 we even get the estimate 1/2 at the right hand side of (4.2), and easily find that k(Ln ALn A−1 Ln )−1 k ≤ 2. Therefore the right inverses of Ln ALn (and also the left inverses) are uniformly bounded. Remark 4.4. The notion of quasi-diagonality is due to Halmos [28] and was initially studied for bounded linear operators A on a separable Hilbert space X. There, an operator A ∈ L(X) was said to be quasi-diagonal, if there was a sequence (Ln ) such that A satisfied (4.1), and the projections (Ln ) were additionally supposed to be compact and orthogonal, and to converge strongly to the identity. In such a setting, Halmos observed that A is quasi-diagonal (in this stricter sense) if and only if A = B + K where B is a block diagonal operator with respect to some orthonormal basis of X ˜ n tending and K is compact. This means that there exist compact and orthogonal projections L ˜ ˜ ˜ ˜ ñ strongly to I which asymptotically commute with A and even fulfill Ln+1 Ln = Ln Ln+1 = L for every n. Berg [3] pointed out that every normal operator is quasi-diagonal in this stricter sense, hence also every self-adjoint operator. So, this class contains a lot of interesting examples like almost Mathieu operators, discretized Hamiltonians, difference operators originated from PDE’s with constant coefficients, certain weighted shifts, etc. We will not go into great detail here, but we 1 This

definition is slightly more general than Definition 1.52.

4.1. QUASI-DIAGONAL OPERATORS

85

refer to the paper of Brown [14] which extensively treats the Hilbert space case, including the discussion of convergence rates for certain classes of quasi-diagonal operators and the definition of some explicit algorithms for two special cases. That paper is heavily based on the results of the Standard Algebra approach in [30], which provides the machinery to describe stability and several asymptotic properties of operator sequences in a Hilbert space setting. Of course, the approach of the present text suggests the expansion to the Banach space case, and this is what we do in the following. The preceding observations reveal that a sequence of projections (Ln ) which meets the requirements given in the definition of a quasi-diagonal operator A are fairly good candidates for the constitution of appropriate finite sections Ln ALn . So, in what follows, we additionally assume that (Ln ) is a sequence of P-compact projections tending P-strongly to the identity, where P = (Pn ) is a uniform approximate identity on X, which equips X with the P-dichotomy. Our aim is to apply the results of the general theory of Parts 2 and 3 and to improve Corollary 4.3. Proposition 4.5. The set QDP,(Ln ) of all operators which are quasi-diagonal w.r.t. (Ln ) is a Banach subalgebra of L(X, P) and contains the ideal K(X, P). If A ∈ QDP,(Ln ) is P-Fredholm and B is a P-regularizer for A then B ∈ QDP,(Ln ) . In particular, QDP,(Ln ) is inverse closed in L(X, P). Proof. Notice that for every fixed m and arbitrary k

k(I − Pn )APm k ≤ k(I − Pn )Lk APm k + k(I − Pn )(I − Lk )APm k

≤ k(I − Pn )Lk APm k + k(I − Pn )A(I − Lk )Pm k + k(I − Pn )[A, I − Lk ]Pm k ≤ k(I − Pn )Lk kkAkCP + (CP + 1)kAkk(I − Lk )Pm k + (CP + 1)CP k[A, Lk ]k,

where the second and third term of the last estimate can be made arbitrarily small by choosing k large, and the first term tends to zero as n → ∞ for every fixed k, hence k(I − Pn )APm k → 0 as n → ∞ (and kPm A(I − Pn )k → 0 by an analogous estimate, too). Thus QDP,(Ln ) is a subset of L(X, P). One easily checks that QDP,(Ln ) is a closed subalgebra. For K ∈ K(X, P) we have kK − Pl KPl k → 0 as l → ∞, and for all l k[Pl KPl , Ln ]k = k(I − Ln )Pl KPl − Pl KPl (I − Ln )k ≤ kPl kkKk(k(I − Ln )Pl k + kPl (I − Ln )k) vanishes for n → ∞. Thus K ∈ QDP,(Ln ) . Let A have a P-regularizer B. Then B ∈ L(X, P) by Theorem 1.14 and further I − AB, I − BA ∈ K(X, P), hence BLn − Ln B = B(Ln A − ALn )B + BLn (I − AB) − (I − BA)Ln B

= B(Ln A − ALn )B + B(Ln − I)(I − AB) − (I − BA)(Ln − I)B,

which also tends to zero in the norm as n → ∞. This finishes the proof. Remark 4.6. At this juncture we want to point out again that the underlying general theory of the Parts 2 and 3 and all proofs also work in the classical setting (without P). More precisely, the subsequent theorems remain true, if (Ln ) is a sequence of compact projections which converge ∗-strongly to the identity. In this case there is no need for an approximate projection P and we can replace L(X, P) by L(X), K(X, P) by K(X), P-compact by compact, P-Fredholm by Fredholm, P-strong convergence by ∗-strong convergence and set BP := 1.

The finite section sequences (Ln ALn ) of quasi-diagonal operators A obviously converge Pstrongly to A. So, let FQD be the smallest closed subalgebra of F containing all sequences {Ln ALn } with A ∈ QDP,(Ln ) , where F is the Banach algebra of all bounded sequences {An } with An ∈ L(im Ln ). Notice that W (A) := P-limn An Ln exists for every sequence A = {An } ∈ FQD .

86


Theorem 4.7. A sequence A = {An } ∈ FQD is Fredholm if and only if W (A) is Fredholm. In this case α(A) = dim ker W (A) and β(A) = dim coker W (A) are splitting numbers and limn ind An = ind W (A). l/r If W (A) is not Fredholm then infinitely many approximation numbers sk (An ) tend to zero as n goes to infinity. A sequence A ∈ FQD is stable if and only if W (A) is invertible. Proof. Let T := {0} and let F T ⊂ F denote the Banach algebra of all sequences A = {An } such that (An Ln ) converges P-strongly. Denote the limit again by W (A). Further, introduce the ideal J T := {{Ln KLn } + {Gn } : K ∈ K(X, P), kGn k → 0}. Then FQD ⊂ F T . The set C := {αI : α ∈ C} is a commutative closed subalgebra of F T , the respective snapshots are multiples of the identity, C/G turns into a C ∗ -algebra (provided with (αI + G)∗ := αI + G) with T as maximal ideal space. With M = T and F0 := I it is clear that C defines a globally localizing setting. For A := {Ln ALn } with A ∈ QDP,(Ln ) we have that W (A) = A, hence this snapshot is liftable and almost commuting with F0 . The coset A + G as well as the lifting A0 + G belong to the commutant of C/G, and it is also obvious that A = A0 , i.e. they are locally equivalent in 0. Consequently, the whole algebra FQD is contained in LT = T T by Proposition 3.8. Now apply Theorem 3.9 to prove that A is J T -Fredholm if and only if W (A) is P-Fredholm, and to derive the asserted criteria for the Fredholm property and the stability. Theorems 2.13, 2.14 and 2.9 provide the rest. Theorem 4.8. Let A = {An } ∈ FQD and assume that BP = limn kLn k = 1. Then −1 lim kAn k = kW (A)k, lim kA−1 k, hence lim cond(An ) = cond(W (A)), n k = k(W (A))

n→∞

n→∞

n→∞

the latter only for stable A. For every 0 < α < < β there is an N0 ∈ Z+ such that Bα (sp W (A)) ⊂ u-lim spN, An ⊂ p-lim spN, An ⊂ Bβ (sp W (A)) n→∞

n→∞

for every N ≥ N0 . If, additionally, X or its dual is complex uniformly convex then, for every > 0 and every N ∈ Z+ , the (N, )-pseudospectra of the operators An converge w.r.t. the Hausdorff distance to the (N, )-pseudospectrum of W (A). Proof. Clearly, this follows from Proposition 3.11 and Corollaries 3.12, 3.19 and 3.20 if C defines a faithful localizing setting, which is almost trivial if we take Remark 3.13 into account. We only note that two functions on T have disjoint support iff at least one of them equals zero. Thus T T /G is a KMS-algebra w.r.t. C/G.

4.2. BAND-DOMINATED OPERATORS ON LP (R, Y )

4.2

87

Band-dominated operators on Lp (R, Y )

Let Y be a Banach space. Here we consider the spaces Lp = Lp (R, Y ), with 1 ≤ p < ∞, of all (equivalence classes of) Lebesgue measurable functions f : R → Y for which the p-th power of kf kY is Lebesgue integrable. By L∞ = L∞ (R, Y ) we further denote the space of all (equivalence classes of) Lebesgue measurable and essentially bounded functions f : R → Y . Note that the Banach space structure is given by pointwise addition and scalar multiplication as well as the usual Lebesgue integral norm (or the essential supremum norm in the case p = ∞). In the same vein, we also define the spaces Lp (R− , Y ) and Lp (R+ , Y ) of functions over the half axes. Let 1 ≤ p ≤ ∞ and let Pn stand for the operator of multiplication by the characteristic function of the interval [−n, n) acting on Lp . Then P := (Pn ) forms a uniform approximate identity and Lp has the P-dichotomy. To prove the latter, we note that in case 1 ≤ p < ∞ the dual space (Lp (R, Y ))∗ can be identified with Lq (R, Y ∗ ) where 1/p + 1/q = 1, i.e. P ∗ is an approximate identity on (Lp )∗ , and the space L∞ is complete w.r.t. P-strong convergence. Now Theorem 1.27 applies. For α ∈ R we consider the operator Uα : Lp → Lp ,

(Uα f )(t) := f (t − α)

of shift by α. Let A ∈ L(Lp , P) and h = (hn ) ⊂ R be a sequence with |hn | → ∞ as n → ∞. The operator Ah is called limit operator of A with respect to h if U−hn AUhn → Ah

P-strongly.

The set σop (A) of all limit operators of A is again called the operator spectrum of A. Finally, an operator A is called rich if every sequence h ⊂ Z whose absolute values tend to infinity has an infinite subsequence g such that Ag exists. One may ask why the definition of rich operators is exclusively based on sequences of integers. The reason is that it would be a much (unnecessarily) stronger condition if all sequences of real numbers shall have a subsequence which leads to a limit operator, and the “set of rich operators” would dramatically shrink. Lindner gave a comprehensive explanation of this phenomenon in [44], Section 3.4.13. Nevertheless, one can replace Z in this definition by any other “two-sided infinite” set of isolated points (cf. Section 4.2.2). Definition 4.9. An operator A ∈ L(Lp , P) is said to be band-dominated if, for every ϕ ∈ BUC, lim sup k[A, ϕt,r I]k = 0.

t→0 r∈R

Let ALp denote the set of all such operators. Notice that this definition is in line with the class of (discrete) band-dominated operators as the third item of Theorem 1.55 shows, and also the treatment of the finite sections is very similar. Nevertheless we want to present a standalone consideration here and avoid to play on the results in the discrete setting which are scattered all over the whole text. In the very end of this section we finish with a short discussion of the connections between the discrete and the continuous case.

4.2.1

Standard finite sections

Set Ln := Pn , further define En := im Ln and let FALp ⊂ F denote the Banach algebra which is generated by all sequences {Ln ALn } with rich A ∈ ALp . Clearly, for every sequence A = {An } in FALp , the limit W (A) := P-lim An Ln n→∞

88


exists. In this way, a first natural snapshot is defined and, as in the discrete case, there are two further points of interest, namely where the operators Ln perform the truncation. More precisely, for a given sequence h = (hn ) of positive integers tending increasingly to infinity we set Ehn := im Lhn , T := {−1, 0, +1}, I 0 := I, I ±1 := χR∓ I and E0 := Lp (R, Y ) E±1 := im I ±1

L0hn := Lhn Eh0n : L(im L0hn ) → L(Ehn ), B 7→ B L±1 hn := U∓hn Lhn U±hn

Eh±1 : L(im L±1 hn ) → L(Ehn ), B 7→ U±hn BU∓hn n for every n. By P t := (Ltn ) uniform approximate identities on Et are given such that the Et have the P t -dichotomy and the sequences (Lthn ) converge P t -strongly to the identities I t on Et . As usual, we let FhT denote the Banach algebra of all bounded sequences {Ahn } of bounded linear operators Ahn ∈ L(Ehn ) for which there exist operators W t {Ahn } ∈ L(Et , P t ) for each t ∈ T such that Eh−t (Ahn )Lthn → W t {Ahn } P t -strongly as n → ∞. n Furthermore, introduce HA as the set of all strictly increasing sequences h of positive integers such that W + (Ah ) and W − (Ah ) exist. Of course, for A := {Ln ALn } with rich A ∈ ALp and a strictly increasing sequence h there is always a subsequence g ∈ HA of h. Since the set RT of all rich sequences in F is a closed algebra (see Theorem 2.26), we see that every A ∈ FALp is rich.

Proposition 4.10. Let A ∈ FALp and g ∈ HA . Then there is a subalgebra C ⊂ F T which defines a faithful globally localizing setting in FgT and which turns Ag into a C-localizable sequence in TgT . In other words: FALp ⊂ T RT . Proof. 2 Firstly, let us prove that σop (B) ⊂ ALp for every B ∈ ALp . Let C ∈ σop (B). From Theorem 1.19 we infer that C ∈ L(Lp , P), but we assume that there is a function ψ ∈ C[−1, 1], a sequence (tn ) ∈ R, tn > 0 and tending to zero, such that C := lim sup k[C, ψtn ,r I]k = lim sup k[Ur CU−r , ψtn I]k > 0. n→∞ r∈R

n→∞ r∈R

Clearly, we can choose a sequence (rn ) such that C = limn k[Urn CU−rn , ψtn I]k, and for all sufficiently large n we fix an integer mn with C/2 < k[Urn CU−rn , ψtn I]Pmn k. Since C is a limit operator of B, we can fix real numbers sn such that C/4 < k[Urn U−sn BUsn U−rn , ψtn I]Pmn k for large n, hence C/4 < supr∈R k[Ur BU−r , ψtn I]k for all large n, contradicting B ∈ ALp . Let g ∈ HA , set i1 := 1 and construct a subsequence i = (ik ) of g such that t (t) t sup{k(A(t) gn − W (Ag ))Lk k, kLk (Agn − W (Ag ))k : t ∈ T, gn ≥ ik }
0, we again define inflated copies by ϕt (x) := ϕ( xt ) and we consider the set C := {{ϕn I} : ϕ ∈ C(R2 )} which belongs to F T . The central snapshot W 0 {ϕn I} of {ϕn I} ∈ C equals ϕ(0)I and, for t ∈ δΩ, we have W t {ϕn I} = ϕ(t)I, since s − ϕ(t) → 0 k(V−nt Rnt ϕn (Rnt )−1 Vnt − ϕ(t)I)Pm k = sup ϕ ρt t − n s∈Bm (0) 4 To

t apply the estimate (4.13) with ψ ≡ 1. check the uniform boundedness of the Rn

96


as n → ∞ for every fixed m, and the dual estimate with Pm multiplied from the left holds as well. Provided with the involution ∗ : {ϕn I + G} 7→ {ϕn I + G}, the set C becomes a commutative C ∗ algebra which is isometrically isomorphic to C(R2 ) and C(B1 (0)) since kϕn Ik = kϕk∞ = kϕ◦ξk∞ for every n and ϕ, hence kϕk∞ = lim supn kϕn Ik = k{ϕn I} + GkF /G . Their maximal ideal spaces can be identified (see Theorem 3.2) and for our purposes it is convenient to think of this space as the compactification R2 of R2 with a sphere of points at infinity and with the Gelfand topology. More precisely, a character on C(R2 ) is either of the form x : f 7→ f (x) with x ∈ R2 , or of the form θ∞ : f 7→ f] ◦ ξ(θ) with θ ∈ δB1 (0). Here we fix M := Ω ⊂ R2 and to every x ∈ Ω \ T we associate t(x) = 0. Now, we have to choose the functions Fx ∈ C(R2 ) which take their values in the interval [0, 1] and are identically 1 in a neighborhood of x ∈ Ω, respectively. We do this in such a way that in case x ∈ δΩ the support of Fx is contained in (Ux ∩ Vx ) \ {0}, and in case x ∈ int Ω the function Fx shall vanish on δΩ. Finally notice that the sequence T = {Tn } belongs to LT : Its central snapshot W 0 (T) is the identity, and for t ∈ δΩ the snapshot W t (T) is the operator of multiplication by the characteristic function χKt0 of the cone Kt0 . The localizability of T is immediate from the definition. Consequently, C defines a partially localizing setting and the results of Section 3.2.1 apply. Also notice that {En±t (ϕn I)} ∈ C for every {ϕn I} ∈ C and every t ∈ T . (4.9)

4.3.2

The main result for localizable sequences

Theorem 4.16. The set LT of all C-localizable sequences is a Banach algebra. For every sequence A = {An } of the form TAT + (I − T) with A ∈ LT the following hold • A is a Fredholm sequence if and only if all of its snapshots are Fredholm. In this case X lim ind An = ind W t (A) (4.10) n→∞

t∈T

and the approximation numbers from the right/left of the An have the α-/β-splitting property with the finite splitting numbers X X α= dim ker W t (A) and β = dim coker W t (A). t∈T

t∈T

• If one of the snapshots W t (A) is not Fredholm then, for each k ∈ N, lim srk (An ) = 0

n→∞

• The norms kAn k and kA−1 n k converge and lim kAn k = max kW t (A)k,

n→∞

t∈T

or

lim slk (An ) = 0.

n→∞

5 t −1 lim kA−1 k. n k = max k(W (A))

n→∞

t∈T

• A is stable if and only if all snapshots are invertible. In this case lim cond(An ) = max kW t (A)k · max k(W t (A))−1 k.

n→∞

t∈T

t∈T

• The (N, )-pseudospectra of the operators An converge w.r.t. the Hausdorff distance to the union of the (N, )-pseudospectra of the snapshots. 5 We

again use the convention kB −1 k = ∞ if B is not invertible.

4.3. MULTI-DIMENSIONAL CONVOLUTION TYPE OPERATORS

97

Proof. Recall from Section 3.2.1 the definition of T T , the set of all sequences being of the form TAT + λ(I − T) + J with A ∈ LT , a complex number λ and a sequence J ∈ J T . Proposition 3.8 provides that LT and T T are Banach algebras and Theorem 3.9 yields the criteria for the stability and the Fredholm property. Then we derive the index formula from Theorem 2.14, the splitting property from Theorem 2.13, and the convergence of the approximation numbers in the case of a non-Fredholm snapshot from Theorem 2.9. We check that C even defines a faithful localizing setting. In the case p = ∞ the estimate k(ϕn An ϕn + ψn Bn ψn )f k = max{kϕn An ϕn f k, kψn Bn ψn f k} ≤ max{kAn k, kBn k}kf k

(4.11)

holds for all f ∈ X, all {An }, {Bn } ∈ F T , all n ∈ N and all ϕ, ψ ∈ C(R2 ) of norm 1 with disjoint support. For 1 ≤ p < ∞ we replace (4.11) by k(ϕn An ϕn + ψn Bn ψn )f kp = kϕn An ϕn f kp + kψn Bn ψn f kp

≤ kϕn An kp kϕn f kp + kψn Bn kp kψn f kp ≤ max{kAn kp , kBn kp }kf kp ,

and we conclude from Remark 3.13 that T T /G is a KMS-algebra with respect to C/G in all cases p ∈ [1, ∞]. We now show that for every A ∈ T T , every g ∈ H+ and every x ∈ Ω kφx (A + G)k ≤ kW x (A)k ≤ kAg + Gk.

(4.12)

We start with the following observation: Let > 0 and t ∈ δΩ be given. Then there is a function ψ ∈ C(R2 ) such that kψk∞ = 1, ψ(t) = 1, Ft ψ = ψ and kψn (Rnt )−1 kkRnt ψn Ik ≤ 1 + for all n. In the case p = ∞ this follows from the fact k(Rnt )−1 k = kRnt k = 1. If 1 ≤ p < ∞ then Z Z s s p k(Rnt ψn )f kp = |(Rnt ψn f )(s)|p ds = f nρt ψn nρt ds n n R2 R2 Z 0 u 0 u = |ψn (u)f (u)|p det(ρ−1 du ≤ sup det(ρ−1 kf kp (4.13) t ) t ) n n u∈supp ψn R2 =

sup

u∈supp ψ

0 p | det(ρ−1 t ) (u)|kf k

−1 0 0 t −1 give together with (ρ−1 t ) (t) = I, the continuity of (ρt ) and an analogous argument for (Rn ) the claim for sufficiently small support of ψ. We conclude for every x ∈ Ω

kφx (A + G)k = kφx (Ax + G)k = kφx ({ψn I}Ax {ψn I} + G)k

≤ k{ψn I}Ax {ψn I} + Gk = lim sup kψn Enx (W x (A))ψn Ik n→∞

≤ lim sup kψn (Rnx )−1 kkVnt(x) kkW x (A)kkV−nt(x) kkRnx ψn Ik ≤ (1 + )kW x (A)k. n→∞

Theorem 1.19 further yields kW x (A)k ≤ lim inf kEg−x (ψgn Agn ψgn )k n n→∞

≤ lim sup kV−gn t(x) kkRgxn ψgn IkkAgn kkψgn (Rgxn )−1 kkVgn t(x) k ≤ (1 + ) lim sup kAgn k. n→∞

n→∞

Since was arbitrary, we arrive at (4.12) and thus, in view of Remark 3.13, we have proved that C defines a faithful localizing setting. Now, Proposition 3.11 provides the convergence of kAn k and kA−1 n k and Corollary 3.12 the convergence of the condition numbers in case A is a stable sequence. From Proposition 3.16 we conclude that all spaces Et or their duals are complex uniformly convex, hence Corollary 3.19 yields the convergence of the pseudospectra. We note that the snapshots At = W t (TAT + (I − T)) are operators acting on the respective “local” cones Kt0 since they are of the form At = χKt0 At χKt0 I + (1 − χKt0 )I.

98


Some members of LT

4.3.3

Since LT is a Banach algebra, we get a whole closed subalgebra of LT and the above results for each of its elements for free, if we only have the generators of this algebra. So, let us start finding interesting individuals in LT . Some basic ingredients We have already seen that T is C-localizable, and that its snapshots W t (T), t ∈ δΩ are the projections χKt0 I, respectively. Also the identity I belongs to LT . Furthermore, all sequences A ∈ J T are C-localizable and have P-compact snapshots. This particularly includes all sequences {J} with a P-compact operator J, and hence all {gI} arising from operators of multiplication by functions g ∈ L∞ (R2 ) with k(I − Tn )gk∞ → 0 as n → ∞. The respective snapshots are W 0 {J} = J, W 0 {gI} = gI, and all other snapshots vanish. Convolution operators Of course, the most challenging operators here are the convolutions C(a) with a ∈ W (R2 ), and we are going to show that {C(a)} is in LT . Actually, we do not need to consider all symbols a = λ + F (u) with λ ∈ C and u ∈ L1 (R2 ) since LT is a Banach algebra containing the identity I, but we can assume that λ = 0 and u is continuous with compact support. For the latter notice that every u ∈ L1 (R2 ) can be approximated by continuous functions having compact support and that kC(F (u))k ≤ kukL1 .6 Proposition 4.17. Let u be continuous with compact support. 1. The commutator [{C(F (u))}, {ϕn I}] belongs to G for every ϕ ∈ C(R2 ). 2. For every > 0 and every t ∈ T there is a function ψ ∈ C(R2 ) of norm one, identically 1 in a neighborhood of t and with small support such that kψn (Ent (C(F (u))) − C(F (u)))k < for sufficiently large n. 3. For every compact set M , both χM C(F (u)) and C(F (u))χM I are compact operators. 7

Proof.

Let the support of u be contained in the ball BR (0). From kC(F (u))k ≤ kukL1 and Z x y (ϕn C(F (u)) − C(F (u))ϕn I)f (x) = u(x − y) ϕ −ϕ f (y)dy, n n BR (x)

using the Hölder inequality, we conclude that kϕn C(F (u)) − C(F (u))ϕn Ik ≤ cn kukL1 , where x y cn := sup sup ϕ −ϕ = sup sup |ϕ (r) − ϕ (s)| . 2 n n x∈R y∈BR (x) r∈R2 s∈B R (r) n

Since |ξ −1 (r) − ξ −1 (s)| ≤ |r − s| we find cn = sup sup ϕ ◦ ξ ξ −1 (r) − ϕ ◦ ξ ξ −1 (s) ≤ sup r∈R2 s∈B R (r) n

sup

a∈U1 (0) b∈B R (a)∩U1 (0)

|ϕ ◦ ξ (a) − ϕ ◦ ξ (b)|

n

and the uniform continuity of ϕ ◦ ξ yields cn → 0 as n → ∞, the first assertion. For the second assertion let t ∈ δΩ and notice that, due to the shift invariance of C(F (u)) and the first part, it suffices to show that kχnW (Rn−t C(F (u))Rnt − C(F (u)))ψn Ik < /2 for every n, where W denotes the support of ψ. Set γ := /(4kχB2R (0) kL1 ). 6 See 7 The

[44], Example 1.45, for instance. proofs are modifications of those in [49] and [72].


99

For f ∈ X and x ∈ nW we find that χnW (Rn−t C(F (u))Rnt − C(F (u)))ψn f (x) equals Z Z s s x − s ψ nρ f nρ ds − χnW (x) u nρ−1 u(x − s)ψ (s)f (s)ds t t n t n n n R2 R2 Z y i y h x 0 = − nρ−1 χnW (x) u nρ−1 det(ρ−1 − u(x − y) ψn (y)f (y)dy. t ) t t n n n nW By Taylor’s Theorem we have −1 ∆ :=n ρ−1 t (v) − ρt (w) − (v − w) −1 −1 0 ≤ n ρt (v) − ρ−1 t (v) − (ρt ) (v) (v − w) + h(v − w)(v − w) − (v − w) 0 ≤ k(ρ−1 t ) (v) − I + h(v − w)kn|v − w|,

with kh(v − w)k → 0 as |v − w| → 0. Fix δ ∈ (0, R). Due to the properties of the diffeomorphism 0 −1 ρt we can choose W sufficiently small such that k(ρ−1 for t ) (v) − I + h(v − w)k ≤ δ(2R) all v, w ∈ W . Then, for every n and v, w ∈ W , ∆ ≤ δ holds in case n|v − w| ≤ 2R, and 0 |n|v − w| − ∆| ≥ R if n|v − w| ≥ 2R. Moreover, det(ρ−1 and equals 1 in t, t ) (·) is continuous 0 that is, by an appropriate choice of W , we can guarantee that 1 − det(ρ−1 ) (·) ≤ γ/(kuk∞ ) t on W . Notice that the continuous function u with a compact support is uniformly continuous. Hence the δ ∈ (0, R) can be fixed such that sup

sup |u(x − y + α) − u(x − y)| ≤ γ.

x,y∈nW |α|≤δ

Puzzling all these parts together we obtain for the expressions i y h x 0 y − nρ−1 En (x, y) := χnW (x) u nρ−1 det(ρ−1 − u(x − y) ψn (y) t ) t t n n n

that En is supported in {(x, y) : x, y ∈ nW, |x − y| ≤ 2R}, and supx,y∈R2 |En (x, y)| ≤ 2γ for every n. Now, in the cases 1 ≤ p < ∞, p Z Z dx E (x, y)f (y)dy kχnW (Rn−t C(F (u))Rnt − C(F (u)))ψn f kp = n nW nW ! !p p Z Z Z Z ≤

nW

B2R (x)

|En (x, y)f (y)|dy

dx ≤ (2γ)p

nW

= (2γ)p kC(χB2R (0) )|f |kp ≤ (2γ)p kχB2R (0) kpL1 kf kp =

B2R (x) p

2

|f (y)|dy

dx

kf kp ,

and if p = ∞, then we estimate kχnW (Rn−t C(F (u))Rnt − C(F (u)))ψn f k by Z Z ess sup En (x, y)f (y)dy ≤ ess sup 2γ|f (y)|dy = 2γkC(χB2R (0) )|f |k ≤ kf k. 2 x∈nW x∈nW nW B2R (x)

The assertion is trivial for t = 0. For the third part let Br (0) be a ball containing M , fix a continuous function ζ of norm one, equal to 1 on the set Br+R (0) and supported in the set B2r+R (0), and set D := ζC(F (u))ζI. Then χM C(F (u)) = χM D and C(F (u))χM I = DχM I, whereas D is an integral operator with a continuous kernel function which is compactly supported. It is well known 8 that such integral operators are compact. 8 For

a detailed proof see [63], Lemma 3.2.4, which works for all p ∈ [1, ∞].

100


Together with (4.9) this is everything that we need to directly check the C-localizability of A: All snapshots of A equal C(F (u)), hence are liftable and their liftings also belong to F T . We only note that, for all t, τ ∈ T , En−τ Ent (C(u)) − C(u) = En−τ Ent (C(u) − C(u)) + En−τ [C(u) − Enτ (C(u))]. The commutation properties and the local equivalence are immediate from the previous proposition and (4.9) as well.

Multiplications by continuous functions For a given function f ∈ C(R2 ) and the sequence A := {f I} it obviously holds that W 0 (A) = f I. If t is in δΩ then we conclude from the definitions that the respective snapshot W t (A) equals (f ◦ ξ)(t/|t|)I. Now it is easy to check that A ∈ LT . Corollary 4.18. The algebra A is embedded in LT by taking A ∈ A as the sequence {A}. Cones Let a finite section domain Ω be given and let K be a cone with vertex at 0 such that A := {χK I} ∈ F T . We prove A ∈ LT . Clearly, W 0 (A) = χK I and, for every x ∈ int Ω, the conditions in the definition of localizable sequences are fulfilled, that is W x (A) are “liftable”, the commutation properties hold and A + G, Ax + G are locally equivalent. The cases t ∈ δΩ ∩ int K with W t (A) = I, and t ∈ δΩ \ K with W t (A) = 0 are obvious as well. So, let t ∈ δΩ ∩ δK. For the description of the snapshot W t (A) we note that there is a uniquely determined half space Ht such that K ∩ Br (t) = Ht ∩ Br (t) for sufficiently small r > 0, and we claim W t (A) = χHt I. The operators Ent (χHt I) are operators of multiplication by the characteristic function of a certain set, and they converge P-strongly, since A ∈ F T . Let At denote the limit. Further, all t (χHt I)) are operators of multiplication by a characteristic function. For all P1 (Ekt (χHt I) − Em sufficiently large k, m, say k, m ≥ N , their norms are less than one, due to the convergence to At , t hence they are zero. Consequently, Pn (Ekt (χHt I) − Em (χHt I)) = 0 for all n and all k, m ≥ nN , t t which implies Pn (Ek (χHt I) − A )) = 0 all n and all k ≥ nN , therefore At must be an operator of multiplication by a characteristic function as well. One now easily checks that At = χHt I, and A proves to be C-localizable. The situation is analogous for cones K with vertex at 0 which comply with the weaker condition A := {χK Tn } ∈ F T . In this case W t (A) = χKt0 χHt I for t ∈ δΩ ∩ δK. Example 4.19. We want to point out at this juncture that sequences arising from the operators in A belong to LT for every finite section domain Ω, whereas the operators of multiplication by the characteristic function of a cone require a suitable choice of the finite section domain and the rectifying diffeomorphisms ρx , at least for the points in δΩ ∩ δK. Here are two examples: 1. Of course, it is rather obvious that Ω = B1 (0) can be provided with diffeomorphims ρt for every t ∈ δΩ which act as the identity on the line through t and the origin, and which make Ω appear locally as a half space in t. Then every cone K with vertex at the origin defines a C-localizable sequence {χK I}. In particular, for every operator A with {A} ∈ LT and its compression AK := χK AχK I + (1 − χK )I to the cone K we get {AK } ∈ LT as well, hence we can apply the finite section method for such operators acting on cones: A := T{AK }T + (I − T). The snapshots at all t “outside” the cone are the identity, for t from the interior of K we obtain W t (A) = χKt0 W t {A}χKt0 I +(1−χKt0 )I, that are operators acting on the respective cones Kt0 which are half spaces in this particular case Ω = B1 (0). Solely the snapshots at the points t ∈ δΩ ∩ δK are operators acting on rectangular cones, namely Kt0 ∩ Ht . To avoid this phenomenon, and to ensure that all snapshots at points t ∈ δΩ become operators on half spaces, one could make another construction as follows:


101

K

K δΩ

δΩ

Figure 4.1: Two possible ways how to treat operators on a cone. 2. For a given cone K we choose Ω such that its boundary δΩ is a smooth curve in every point t ∈ δΩ ∩ int K, merges into the boundary δK of K, and is a circular arc outside K (see Figure 4.1). The diffeomorphisms can be chosen in such a way that {χK Tn } ∈ F T . Now, for an operator A with {A} ∈ LT the compression AK := χK AχK I + (1 − χK )I has a C-localizable finite section sequence T{AK }T − (I − T) whose snapshots are either trivial or operators on half spaces for every t ∈ δΩ. We particularly note again that W t {χK Tn } = χKt0 χHt I for t ∈ δΩ ∩ δK. The advantage of the latter example lies in the fact that for convolutions on half spaces the invertibility can effectively be checked. Proposition 4.20. Let a be in the Wiener algebra W (R2 ), H, H 0 be half spaces and K = H ∩ H 0 be a cone. Set A := C(a) as well as AH := χH AχH I +(1−χH )I and AK := χK AχK I +(1−χK )I. Then the following are equivalent: a is invertible; A is invertible; A is Fredholm; A is P-Fredholm; AH is invertible; AH is Fredholm; AH is P-Fredholm; AK is Fredholm; AK is P-Fredholm. Proof. It is a well known fact that the Fredholmness of AK and A are both equivalent to the invertibility of the symbol a. Its proof goes back to Simonenko [85] and can also be found in [12], 9.57. (for 1 ≤ p < ∞, and for p = ∞ by duality and the help of Proposition 1.34). It is also clear that the Fredholmness of AK yields that it is properly P-Fredholm, hence it implies its P-Fredholmness. So, let AK be P-Fredholm, choose a non-zero α ∈ δK and consider P-limn V−nα AK Vnα . Due to the shift invariance of A this limit exists and equals AH . Further notice that this limit is invertible by the discussion after Theorem 1.58. The implications (AH invertible ⇒ AH Fredholm ⇒ AH P-Fredholm) are obvious, and by passing to the P-strong limit P-limn V−nβ AH Vnβ with β ∈ int H we find in the same way that (AH P-Fredholm ⇒ A invertible ⇒ A Fredholm ⇒ A P-Fredholm) holds. Employing this argument a third time we finally get that A P-Fredholm provides itself as an invertible limit, thus (A P-Fredholm ⇒ A invertible ⇒ a invertible ⇒ AK Fredholm). On some simplifications and side effects of Theorem 4.16 Let B0 be the (non-closed) algebra of all finite sums of products of convolution operators C(a) with a ∈ W (R2 ), multiplications f I by f ∈ C(R2 ), compact operators J ∈ L(X, P) and multiplications gI by functions g ∈ L∞ (R2 ) for which k(I − Tn )gk →n→∞ 0 holds. Further let B denote the closure of B0 in L(X), which obviously includes A. The following observation has also been made in [50].

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Corollary 4.21. Let A ∈ B0 , AK its compression to a cone K with vertex at 0, and Ω be chosen in such a way that all snapshots W t {χK Tn }, t ∈ δΩ, are either trivial or operators of multiplication by the characteristic function of a half space. Then, for A := T{AK }T + (I − T), • A is Fredholm iff AK is Fredholm. In this case α(A) = β(A) = dim ker AK and ind AK = 0. • A is stable iff AK is invertible. Proof. Let AK be Fredholm. We show that all snapshots At := W t (A), t ∈ δΩ, are invertible. Then, since A ∈ LT , Theorem 4.16 provides the asserted criteria for the Fredholm property and stability, as well as formulas on the α- and β-number and the index. The snapshots B t := W t {A} with t ∈ δΩ are invertible by analogous arguments as in the discussion after Theorem 1.58. Further it is easy to see that, for t ∈ δΩ \ K, we have At = I. If t ∈ δΩ ∩ K, then the B t are of the form C(at ) with some at ∈ W (R2 ), thus every At (which is the compression of B t to the respective half space Kt0 ) is invertible as well by the previous proposition. Moreover, Theorem 4.16 tells us that, for large n, An is a Fredholm operator. Note that An consists of compact operators, operators of multiplication by certain functions (which commute with Tn ) and convolutions C(a), a = F (u), u ∈ L1 . From Proposition 4.17 we conclude that Tn C(a) and C(a)Tn are compact, hence An = fn I+Cn with an operator fn I of multiplication and a compact operator Cn . Consequently, ind An = 0, which specifies the index formula (4.10). Corollary 4.22. Let A ∈ B and AK be its compression to a cone K with vertex at 0. If AK is Fredholm then its index is zero. Proof. By Theorem 1.2 we can choose an operator B ∈ B0 in a neighborhood of A such that its compression BK is Fredholm of the same index as AK . Furthermore, we can choose a finite section domain Ω with the respective diffeomorphisms as in the second part of Example 4.19. Then we are in a position to apply the previous corollary to find ind BK = 0. Finally, as another example, consider a finite section domain Ω being a polygon and all diffeomorphisms ρt being the identical mappings. Define the algebra E T ⊂ LT by the generators {C(a)} with a ∈ W (R2 ), {f I} with f ∈ C(R2 ) and f ◦ ξ being constant on the unit sphere δB1 (0), {J} with compact J ∈ L(X, P), {gI} where g ∈ L∞ (R2 ) with k(I − Tn )gk →n→∞ 0, {χK I} with cones K which intersect δΩ solely in its vertices, and by the sequence T. Then, for a sequence A = TAT + (I − T) ∈ E T , the snapshots W t (A) for all t in the relative interior of one edge of δΩ coincide. Moreover, if the snapshot W v (A) at a vertex v of Ω is Fredholm then the snapshots W t (A) for all t in the interior of the flanking edges can be obtained as P-strong limits of W v (A), hence are invertible. Further, by Theorem 1.19, they fulfill kW t (A)k ≤ kW v (A)k and k(W t (A))−1 k ≤ k(W v (A))−1 k. For the latter we again refer to the proof of Theorem 1.58. Thus, in this particular setting, we can replace the index set T in Theorem 4.16 by the finite set which consists of 0 and the vertices of Ω. This was also discussed in [50] and picks up results of Maximenko [51].

4.3.4

Rich sequences

The consideration of rich sequences in the present text permits to include some more classes of multiplication operators which require the passage to subsequences, but then provide convenient snapshots again. Here we consider the set RLT ⊃ LT of all sequences A ∈ F with the property


103

that every subsequence of A has a Cg -localizable subsequence Ag , that is Ag ∈ LTg . Moreover, we let RT T denote the set of all sequences TAT + λ(I − T) + J with A ∈ RLT , λ ∈ C and J ∈ J T . Note that, in contrast to the set T RT as introduced in Section 3.2.4, we now have one common localizing setting for all subsequences under consideration. Of course, these definitions are much more restrictive, but their greatest advantage is that RLT and RT T can straightforwardly be proved to be Banach algebras. Therefore we again only need to investigate some generators. Nevertheless, RT T is included in the set T RT , hence the main results of Theorem 3.22 are applicable to all sequences in RT T . Thus, for the Fredholm and stability criteria, the α- and β-numbers and for the formulas which describe the asymptotics of the norms, condition numbers and pseudospectra of each sequence A ∈ RT T we refer to Theorem 3.22 without stating the results again. Here we only get to know some further elements in RLT . Very slowly oscillating functions We say that a bounded and continuous function f is very slowly oscillating, if for each x 6= 0 and each > 0 there is a δ > 0 such that |f (rs) − f (rx)| < for all s ∈ Uδ (x) and all r > 0. The set VSO of all very slowly oscillating functions is a Banach algebra and includes C(R2 ). A basic example for a function f ∈ VSO \C(R2 ) is given by f (x) = sin ln |x|, |x| ≥ 1. Proposition 4.23. For every f ∈ VSO the sequence A = {f I} belongs to RLT with W 0 (A) = f I and the other snapshots W t (Ag ) at the boundary points t ∈ δΩ are multiples of the identity. Proof. Clearly, if we have a T -structured subsequence Ag of A then it follows immediately from the definition that Ag belongs to LTg with the snapshots as asserted. Therefore, it suffices to show that A is rich. Let h ∈ H+ , and choose a dense and countable subset {ti }i∈N of T . Then there is a subsequence g 1 ⊂ h such that W t1 (Ag1 ) exists. To see this, notice that the sequence (f (hn t1 ))n∈N has a convergent subsequence (f (gn1 t1 )n∈N with limit ft1 due to the Bolzano-Weierstrass Theorem. Now one easily derives from the definition of VSO that W t (Ag1 ) = ft1 I. By this argument we successively find subsequences g j+1 ⊂ g j such that W tj+1 (Agj+1 ) exists. Define gn := gnn for every n ∈ N and find that W s (Ag ) exists for every s in this dense subset {ti }i∈N of T . For an arbitrary t ∈ T there is a sequence (sm )m ⊂ {ti }i∈N which converges to t. One now easily checks that (W sm (Ag ))m converges to an operator B and that B provides the P-strong limit of (Eg−t (f I)). Thus, Ag ∈ FgT . n We note that the snapshots of VSO-sequences are shift invariant, hence the above Corollaries 4.21 and 4.22 stay valid if we enrich the algebras B0 and B by very slowly oscillating functions. Also the algebra E T permits an extension in this vein. Moreover, we want to point out that the sequences arising from VSO are treatable by arbitrary finite section domains Ω. Finally, we consider a further class of multiplications which depends on the shape of Ω. Periodic functions Let the finite section domain Ω be given. A function f ∈ C(R2 ) is said to be Ω-periodic (or periodic with respect to Ω) with period P if, for every t ∈ δΩ, the functions ft : [1, ∞) → C, α 7→ f (αt) are periodic with period P , that is ft (α) = ft (α + P ) for every α. A simple example for such a function is shown in Figure 4.2, where Ω = B1 (0) and P equals 2π 7 . Let f be Ω-periodic, h be a strictly increasing sequence of positive integers, and decompose each hn = kn P +rn +1 with kn ∈ Z and rn ∈ [0, P ). Due to the Bolzano-Weierstrass theorem there is a

104


1 0 3

−1 3

2 2

1 1 0 0 −1 −1 −2

−2 −3

−3

Figure 4.2: The function f (z) := sin(7|z|) sin(5 arg(z)), z 6= 0 is Ω-periodic for Ω = B1 (0). convergent subsequence of (rn ) with limit r, and we denote by g the respective subsequence of h. Then, for every t ∈ δΩ, the operator W t (Ag ) exists and equals to the operator of multiplication by the continuous function which takes the value f ((1Student + Version r +ofs)t) on the set δKt0 + nP + s for MATLAB every n ∈ Z and s ∈ [0, P ). If, in particular, for one t ∈ δΩ, the boundary δKt of the local cone Kt contains the origin then f proves to be constant along the ray αt, α ∈ [1, ∞), hence W t (Ag ) = f (t)I. Now, one easily verifies Proposition 4.24. For every Ω-periodic function f the sequence A = {f I} belongs to RLT . Remark 4.25. We finally want to mention, without giving details, that bounded and continuous functions f which are constant along each curve δ(αΩ), α > 0, and which are slowly oscillating also define operators A = f I of multiplication that can be tackled by the tools of the present text. Here and in the literature a continuous function is called slowly oscillating if, for every r > 0, sup{|f (x) − f (y)| : y ∈ Ur (x)} → 0 as |x| → ∞.

The idea is to check that the arising sequences A ∈ F are rich, to construct an adapted “blowingup-procedure” and to define a localizing setting as in Proposition 4.10 such that one finally gets A ∈ LRT . The present ideas and proofs for convolution type operators on Lp (R2 ) also work for analogous operators on Lp (RN ) and even on Lp (RN , Cm ), the product of m copies of the space Lp (RN ). The discrete multidimensional setting lp (ZN , X) is more involved, since it is not possible to rectify the boundary δΩ by diffeomorphisms as in the continuous case. At least for finite sections with respect to polyhedra Ω which have their vertices in ZN there are comprehensive results [62], [60], Section 6.2 in [63], [69] for the case p = 2, and Section 4.1 in [44]. For deeper results on the Fredholm theory for such finite section sequences and its consequences we refer also to [69], which considers the case p = 2 and already provides results on the asymptotics of norms, condition numbers and pseudospectra. An extension of the Fredholm theory to the cases p ∈ [1, ∞] with finite dimensional X can be found in [80]. By the tools of the present text this now can easily be done for arbitrary Banach spaces X, and can be completed by the statements on the norms, condition numbers and pseudospectra, by the index formula and by certain further classes of rich sequences. Notice that Lindner [43] also considered the stability of finite section sequences with respect to more general starlike domains Ω.

4.4. FURTHER RELATIVES OF BAND-DOMINATED OPERATORS

4.4

105

Further relatives of band-dominated operators

At the end of the 4th part we return to our prototypic example, the class of band-dominated operators on lp (Z, X), which has been a guide through the whole text, and we touch upon some related families of operators. At that we have two different aims in mind. On the one hand we want to translate what we obtained in the already studied case to these similar settings. We weaken the notion of band-dominated operators into several directions without loss of the essential results on the applicability of the finite section method. Just to give some keywords, we deal with weakly band-dominated, triangular, quasi-triangular and triangle-dominated operators. On the other hand, for more specific families of operators, such as some comfortable classes of Toeplitz operators or slowly oscillating operators, also the conditions in the theorems and the respective conclusions turn into a simpler and more concrete form which we want to expose. Since this is just to give some links to well known forerunners and to sound out some possible future work, we keep the presentation as brief as possible.

4.4.1

Band-dominated operators on lp (N, X)

Recall the approximate projection P and the sequence (Ln ) in L(lp (Z, X)). Let P ∈ L(lp (Z, X)) denote the canonical projection onto the subspace of all sequences (xi ) whose entries xi vanish for i ≤ 0. This subspace can be (isometrically) identified with the space lp (N, X) of all one-sided infinite and p-summable (for 1 ≤ p < ∞) or bounded (for p = ∞, resp.) sequences. For every bounded linear operator A on lp (Z, X) the compression P AP yields a bounded linear operator on lp (N, X), and vice versa for every B ∈ L(lp (N, X)) we have P BP + Q ∈ L(lp (Z, X)), where Q := I − P . We introduce the algebra Alp (N) of band-dominated operators on lp (N, X) as the set of all compressions of band-dominated operators on lp (Z, X), and note that we would obtain exactly the same if we would build upon shifts and operators of multiplication by bounded sequences as generators for the algebra Alp (N) . Analogously, we take the compressions P Ln P as the finite section projections, denote them again by Ln , and let FAlp (N) stand for the Banach algebra which is generated by all finite section sequences of rich band-dominated operators. Then we can completely translate Theorems 2.31, 2.32, 2.33, 2.35 and 3.23 in the sense of this mentioned embedding. Of course, for every A ∈ FAlp (N) , we have the basic snapshot W 0 (A) which appears in the form P W 0 (A)P + λQ, and only one further “direction” which provides relevant snapshots W 1 (Ag ) since the snapshots W −1 (Ag ) are always trivial. One usually applies the flip operator J, given by (xi ) 7→ (x1−i ), to achieve that these W 1 (Ag ) can be regarded as operators JW 1 (Ag )J acting on lp (N, X) as well. Instead of using such flipped versions JEn−1 (An )L1n J = JVn An Ln V−n J of En−1 (An ) for the construction of the snapshots at t = 1, one can also employ the operators Wn : lp (N, X) → lp (N, X), (x1 , x2 , x3 , . . .) 7→ (xn , xn−1 , . . . , x1 , 0, 0, . . .) and set En−1 (An ) := Wn An Wn to obtain exactly the same outcome. Actually, the second approach has been the initial one [83] and constitutes a cornerstone in the history of the Banach algebra techniques which make up the largest part of this thesis. Before we turn to the class of Toeplitz operators that has been the engine in the development throughout the last decades we shortly examine a class of band-dominated operators on lp (N, C) whose finite sections possess surprisingly simple Fredholm and stability criteria.

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4.4.2


Slowly oscillating operators

We call a function a ∈ l∞ (N, C) slowly oscillating if, for every r > 0, sup{|a(x) − a(y)| : y ∈ N such that |x − y| ≤ r} → 0 as |x| → ∞. A band-dominated operator A is said to be a slowly oscillating operator if its coefficients are slowly oscillating, that is A is the norm limit of a sequence of band operators having only slowly oscillating coefficients. Theorem 4.26. Let A be a slowly oscillating operator and A := {Ln ALn } be the sequence of its finite sections. Then A is Fredholm if and only if A is a Fredholm operator. In this case α(A) = β(A) = max{dim ker A, dim coker A} are splitting numbers. In particular, the invertibility of A is necessary and sufficient for the stability of A. This result on the stability has been established by Roch, Lindner and Rabinovich in [46] (p = 2, banded operators) and [68] (1 < p < ∞) for slowly oscillating operators, whereas it is well known for a much longer time for large classes of Toeplitz operators (see the pioneering work in [24], [83] or the comprehensive monographs [12], [13]). For the remaining cases p = 1, ∞ Theorem 4.26 is new. Actually, the additional snapshots of the sequences in this Theorem are Toeplitz, and the proof immediately follows if we invest some facts on such operators. Therefore we defer the proof to the end of the next section.

4.4.3

Toeplitz operators with continuous symbol

A one-sided infinite matrix T (a) := (aj−k )j,k∈N with complex entries ai is called Toeplitz matrix. P Suppose that T (a) induces a bounded linear operator on lp (N, C) via (xi ) 7→ ( k∈N ai−k xk ). Then there is a function a ∈ L∞ (T) on the unit circle T whose Fourier coefficients form the sequence (ai )i∈Z , we further denote this operator again by T (a) and call it the Toeplitz operator with the generating function (or symbol function) a. The set of all such symbols a which define a Toeplitz operator T (a) shall be denoted by M p in what follows, and it is well known that, provided with pointwise operations and the norm kakM p := kT (a)kL(lp (N,C)) , M p forms a Banach algebra, the so-called multiplier algebra. Recall the shift operators Vk and the finite section projections Ln on lp (Z, C), as well as the canonical projection P which maps lp (Z, C) onto lp (N, C) and denote the compressions P Vk P and P Ln P again by Vk and Ln . Clearly, the shift Vk on lp (N, C) is a Toeplitz operator having the trigonometric polynomial t 7→ tk as generating function. So, we see that for every trigonometric polynomial a the corresponding Toeplitz operator T (a) is banded. Let C p denote the closure of the set of all trigonometric polynomials in M p and refer to its elements as continuous symbols. This definition immediately gives that every Toeplitz operator with continuous symbol is band-dominated. It is well known that M 2 = L∞ (T) and C 2 = C(T), but for p 6= 2 the equalities must be replaced by proper inclusions “⊂”. Moreover, M 1 = C 1 = M ∞ = C ∞ = W is the P so-called Wiener algebra W of all functions a ∈ L∞ (T) whose Fourier coefficients (ai ) fulfill i∈Z |ai | < ∞. All these definitions and results are taken from [12], Chapter 2 for the cases 1 ≤ p < ∞. The case p = ∞ is tackled by its duality to l1 and the observation T (a)∗ = T (a) in [8], for example. Theorem 4.27. (cf. [12], 2.47. and Coburns theorem [12], 2.38.) Let a ∈ C p . Then the Toeplitz operator T (a) is Fredholm if and only if a does not have any zeros on T. In this case, T (a) is one-sided invertible and its index equals − wind(a, 0), that is minus the winding number of the curve a around the origin. Moreover, a−1 ∈ C p and T (a−1 ) is a regularizer for T (a).


107

Theorem 4.28. Let A = {An } := {Ln T (a)Ln } with a ∈ C p . Then • A is Fredholm if and only if a does not vanish on T. In this case the approximation numbers of the An have the splitting property with α(A) = β(A) = | ind T (a)| = max{dim ker T (a), dim coker T (a)} = | wind(a, 0)|. Otherwise, all approximation numbers tend to zero as n → ∞. • The norms kAn k and kA−1 ˜(z) := a(z −1 )) n k converge to the limits (with a max{kT (a)k, kT (˜ a)k}

and

max{k(T (a))−1 k, k(T (˜ a))−1 k},

respectively.

• For every > 0 and every N ∈ Z+ the (N, )-pseudospectra spN, An of the finite sections converge with respect to the Hausdorff distance to the union spN, T (a) ∪ spN, T (˜ a) as n goes to infinity. • A is stable if and only if a does not vanish on T and its winding number is zero. In this case the condition numbers cond(An ) converge to max{kT (a)k, kT (˜ a)k} · max{k(T (a))−1 k, k(T (˜ a))−1 k}.9 This theorem has grown to several expansion stages since the middle of the 20th century, and it holds even for much larger classes of Toeplitz operators, such as for piecewise continuous symbols. A beautiful treatise on this topic with a clear and standalone presentation of the theory around Toeplitz operators and its history can be found in the already mentioned monograph [13] by Böttcher and Silbermann. The particular case of banded Toeplitz matrices is subject of the book [8] by Böttcher and Grudsky. For the most complete resource on this topic and its development we refer to the comprehensive monograph [12] by Böttcher and Silbermann. In the present work, Theorem 4.28 is just a special case of what we know for band-dominated operators on lp (N, C) from Section 4.4.1 and Theorem 4.26, taking Theorem 4.27 into account. We also want to record that the smallest closed subalgebra of L(lp (N, C)) containing all Toeplitz operators with continuous symbol is of the form {T (a) + K : a ∈ C p , K P-compact}, and the smallest closed subalgebra of F containing all respective finite section sequences equals {{Ln T (a)Ln + Ln KLn + Wn LWn + Gn } : a ∈ C p , K, L P-compact, {Gn } ∈ G}. To check this take [12], 2.14, 7.7, and Proposition 1.20 into account. We close this section with the pending Proof of Theorem 4.26. Let A be a slowly oscillating Fredholm operator and A = {Ln ALn } be its finite section sequence. As in Section 4.4.1 we see that Corollary 2.30 can be translated to the lp (N, C)-case, hence provides that every snapshot of A is Fredholm. Notice that A is rich, due to Corollary 1.60. Let g ∈ H+ be such that W 1 (Ag ) exists, and choose a sequence of banded slowly oscillating operators Am which converge to A in the norm as m goes to infinity. Now, pass to a subsequence h1 of g such that W 1 {Lh1n A1 Lh1n } exists, then to a subsequence h2 of h1 such that W 1 {Lh2n A2 Lh2n } exists, and so on. Then define h = (hn ) by hn := hnn . It is obvious from 9 The

additional operators T (˜ a) cannot be omitted in general, see [13], Example 7.14 for instance.

108


the definition that the snapshots W 1 {Lhn Ak Lhn } are banded Toeplitz operators for every k and converge to W 1 (Ah ), which equals W 1 (Ag ), as k goes to infinity hence W 1 (Ag ) is a Fredholm Toeplitz operator with a continuous generating function. Moreover, from Theorem 2.31 we get that 0 = limn ind Lhn ALhn = ind A + ind W 1 (Ag ). Therefore dim ker A + dim ker W 1 (Ag ) = dim coker A + dim coker W 1 (Ag ) where, by Theorem 4.27, at least one of these four non-negative integers is equal to zero. Employing Theorem 2.31 again, we find the asserted splitting numbers at least for all subsequences Ag which have a snapshot W 1 (Ag ), and Theorem 2.32 provides the Fredholm property of A and the α- and β-number of the whole sequence A. Assume that α(A) is not a splitting number for A. Then there is a subsequence Aj of A such that limn srα(A) (Ajn ) exists and is larger than zero. Now pass to a subsequence Ag of Aj which has a snapshot W 1 (Ag ) and gives a contradiction. The same argument works for β(A). The stability criterion follows from Theorem 2.21.

4.4.4

Cross-dominated operators and the flip

Here, we consider the closed algebra Xlp of bounded linear operators on lp (Z, X) which is generated by the shift operators Vα , α ∈ Z, by the operators of multiplication aI by sequences a ∈ l∞ (Z, L(X)) and, in contrast to the algebra Alp of band-dominated operators, additionally by the flip J which maps the sequence (xi ) to (x1−i ). We refer to such operators as crossdominated operators, which is reasonable since the matrix representations of finite combinations of the generators look like a “cross” consisting of two orthogonal bands of finite width. For some similar material on an algebra of convolution type operators and a flip we refer to [71]. Clearly, the approach of Section 2.5 is not appropriate here, since the homomorphisms En±1 as defined there do not provide a P ±1 -strongly convergent copy (En±1 (An )L±1 n ) of the finite section sequence {An } := {Ln JLn } associated to the flip operator, not even for any subsequence. Hence, one gets the impression that the present setting is more involved. Actually, this is false. To see this, we use a tricky transformation which shuffles the entries of the sequences under consideration: ( if k is even xk 2 S : lp (Z, X) → lp (N, X), (xi )i∈Z 7→ (yk )k∈N with yk = x 1−k if k is odd. 2

Obviously, this is an isometric isomorphism, and it can now straightforwardly be checked, by considering the generators, that Ψ : A 7→ SAS −1 provides an isometric algebra isomorphism which translates Xlp into the algebra Alp (N) of band-dominated operators on lp (N, X). Also the finite section projections Ln are transformed into the respective finite section projections ˜ n : lp (N, X) → lp (N, X) : (yi )i∈N 7→ (y1 , y2 , . . . , y2n+1 , 0, . . .). L Thus, all results on the Fredholm property, on stability, on the formulas for the α- and β-numbers and the index, on the asymptotic behavior of norms, condition numbers and on the convergence of pseudospectra are available for cross-dominated operators as well. To be a bit more precise we note that the isometry Ψ translates the operator of multiplication by the sequence (. . . , a−2 , a−1 , a0 , a1 , a2 , . . .) ∈ l∞ (Z, L(X)) into the operator of multiplication by the bounded sequence (a0 , a1 , a−1 , a2 , a−2 , . . .) ∈ l∞ (N, L(X)), and vice versa. Furthermore, ˆ 1 := P L1 P , we straightforwardly check that using dˆ := (I, 0, I, 0, I, 0, . . .), Vˆk := P Vk P and L ˆ Vˆ2 + Vˆ1 L ˆ1 Ψ(V1 ) = SV1 S −1 = dˆVˆ−2 + (I − d)

ˆ and Ψ(J) = SJS −1 = dˆVˆ−1 + Vˆ1 (I − dI).


In matrix notation this  0 I  0   Ψ(V1 ) =     

is 0 0 0 I

I 0 0 0 0

0 0 0 0 I

I 0 0 0

 0 0 0 ..

Conversely, Ψ−1 (Vˆ1 ) = V−1 JP + J(I − P )  . ..   I 0   I 0   0 0   0 I   0 I   I  . . .

4.4.5

109

.

     ,    



0 I     Ψ(J) =     

I 0 0

0 0 I

I 0 0

 0 0 I

I 0 ..

.

     .    

and Ψ−1 (Vˆ−1 ) = JP + P V−1 J(I − P ), i.e.    .. .       0 I       0 I       0 I ,  .    I 0       I 0       0    . ..

Quasi-triangular operators

Definition 4.29. Let X be a Banach space. Let further A ∈ L(X) and assume that there is a bounded sequence (Ln ) ⊂ L(X) of projections, such that k(I − Ln )ALn k → 0 as n → ∞. Then A is said to be quasi-triangular with respect to (Ln ). If even (I − Ln )ALn = 0 holds for every n then A is called triangular w.r.t. (Ln ). For a collection of several basic properties and applications of quasi-triangular operators on Hilbert spaces see [2]. By some straightforward adaptions we find that analogues of Proposition 4.2 and Corollary 4.3 hold for an invertible quasi-triangular operator A whose inverse is quasi-triangular, too. So, we again see that a sequence of projections (Ln ) which makes an operator A quasi-triangular is quite well suited to define a finite section method. In what follows, we assume that (Ln ) itself converges in a convenient manner: Let P = (Pn ) be a uniform approximate identity on X, which equips X with the P-dichotomy, and suppose that all Ln are P-compact projections which tend P-strongly to the identity. (We do not grow tired of noting that the classical situation with compact Ln which converge ∗-strongly, and without a P, is covered by our proofs as well.) Proposition 4.30. The set QTP,(Ln ) of all operators in L(X, P) which are quasi-triangular w.r.t. (Ln ) is a Banach subalgebra of L(X, P) and contains QDP,(Ln ) the set of all quasi-diagonal operators (w.r.t. (Ln )) as well as the ideal K(X, P). The proof of this proposition is easy. Furthermore the finite section sequence (Ln ALn ) of each operator A ∈ QTP,(Ln ) converges P-strongly to A since A ∈ L(X, P) and Ln → I P-strongly. So, we introduce FQT as usual, namely as the closed algebra which is generated by all sequences {Ln ALn } with A ∈ QTP,(Ln ) . Notice that W (A) := P-limn An Ln exists for every sequence A = {An } ∈ FQT by Theorem 1.19, and that W (A) is quasi-triangular.

110


Theorem 4.31. Let A = {An } ∈ FQT and suppose that W (A) is Fredholm with a regularizer in QTP,(Ln ) . Then the sequence A is Fredholm, α(A) = dim ker W (A) and β(A) = dim coker W (A) are splitting numbers and limn ind An = ind W (A). If W (A) is not Fredholm then infinitely many approximation numbers srk (An ) or slk (An ) tend to zero as n → ∞. The sequence A is stable if W (A) is invertible and its inverse belongs to QTP,(Ln ) . It is not stable if W (A) is not invertible. Proof. We take up the sequence algebraic framework F T as defined for quasi-diagonal operators in Section 4.1 and note that A ∈ F T . Firstly, we check that A − {Ln W (A)Ln } ∈ G: This obviously holds true if A is the finite section sequence {Ln ALn } of a single A ∈ QTP,(Ln ) , hence also for a finite product of the form Πi {Ln Ai Ln } with Ai ∈ QTP,(Ln ) (where we use that k(I − Ln )Ai Ln k → 0), and since G is a closed ideal the assertion follows for all A ∈ FQT . Now, let B ∈ QTP,(Ln ) be a regularizer for A := W (A). Then B := {Ln BLn } belongs to F T as well and An Ln BLn = Ln ABLn − Ln A(I − Ln )BLn + (An − Ln ALn )Ln BLn

= Ln + Ln (AB − I)Ln − Ln A(I − Ln )BLn + (An − Ln ALn )Ln BLn

where {Ln A(I − Ln )BLn }, {(An − Ln ALn )Ln BLn } ∈ G and AB − I is P-compact. We can make an analogous observation for Ln BLn An using the quasi-triangularity of A and we get that A is regularly J T -Fredholm, hence Corollary 2.24 together with Theorems 2.13, 2.14, 2.9 and 2.21 give the assertion. Notice that the additional condition - the regularizer belongs to QTP,(Ln ) - is not redundant in general, as a simple example demonstrates: The shift operator V−1 on lp (N, C) is triangular with respect to the standard sequence (Ln ), but its regularizer V1 is neither triangular nor quasitriangular w.r.t. (Ln ). Since all regularizers coincide modulo the ideal of compact operators none of them is quasi-triangular. Even for the case of an invertible operator A in a setting with projections Ln having infinite rank there are counterexamples: Set X = Lp [0, 2] and consider the operators C, D on X given by ( ( f (2x) if x ∈ [0, 1] f (2x − 2) if x ∈ [1, 2] (Cf )(x) = and (Df )(x) = . 0 otherwise 0 otherwise Clearly, both of them are invertible from the left with the left inverses (C (−1) g)(x) = g(x/2) and (D(−1) g)(x) = g(x/2 + 1), respectively. One easily checks that the following operator B on X 3 is invertible   (−1)   C D 0 C 0 0 B =  0 0 C (−1)  and B −1 = D(−1) 0 0  . 0 0 D(−1) 0 C D Now take A ∈ L(lp (Z, X)) as the block diagonal operator having the blocks B along its diagonal. Clearly, A is invertible and triangular w.r.t. (Ln ), but its inverse isn’t. Nevertheless, if all Ln are finite rank operators, A is quasi-triangular w.r.t. (Ln ) and invertible, then A−1 is also quasi-triangular. To see this, note that kLn − Ln A−1 Ln ALn k ≤ kLn kkA−1 kk(I − Ln )ALn k → 0 as n → ∞, that is, for large n, we get the invertibility of Ln ALn from the left, and there is a uniformly bounded sequence of left inverses in L(im Ln ). Since dim im Ln < ∞ the Ln ALn are even


111

invertible with uniformly bounded inverses, hence kLn − Ln ALn A−1 Ln k = kLn ALn (Ln − Ln A−1 Ln ALn )(Ln ALn )−1 Ln k

≤ kLn ALn kkLn − Ln A−1 Ln ALn kk(Ln ALn )−1 Ln k → 0.

Thus, A−1 ∈ QTP,(Ln ) follows from k(I − Ln )A−1 Ln k ≤ kA−1 kkA(I − Ln )A−1 Ln k = kA−1 kkLn − ALn A−1 Ln k

≤ kA−1 kkLn − Ln ALn A−1 Ln k + kA−1 kk(I − Ln )ALn kkA−1 Ln k → 0.

4.4.6

Triangle-dominated and weakly band-dominated operators

Encouraged by the latest observations we ask if we can enlarge the class of band-dominated operators on lp (Z, X) in such a way, that also quasi-triangular operators are included, and (at least some of) the results on the stability or almost stability of the respective finite sections still hold. So, take the usual finite section projections (Ln ) of Section 2.5 for the following definition. Definition 4.32. Let Tlp denote the subset of L(lp , P) whose elements A additionally fulfill lim sup k(I − Ln )ALn−m k = 0.

m→∞ n>m

(4.14)

The operators in Tlp are called triangle-dominated (with respect to (Ln )). Clearly, all band-dominated operators (by Theorem 1.55) as well as all operators which are quasi-triangular w.r.t. (Ln ) are contained in Tlp , and Tlp easily proves to be a Banach algebra. We know that the band-dominated operators require at least three “directions” from which we should look at their finite section sequences to get enough information: Besides the central snapshot W 0 (A) we need to observe the two critical points where the Ln perform the truncation. Therefore, we let FTlp denote the smallest Banach algebra which contains all finite section sequences of rich triangle-dominated operators, and we use an algebraic framework F T as in Section 2.5.1 in order to study these sequences. Notice that W 0 (A) is triangle-dominated for every A ∈ FTlp and, proceeding exactly as in the proof of Proposition 2.28, we find

Proposition 4.33. Let A ∈ FTlp be such that W 0 (A) is P-Fredholm and has a P-regularizer in Tlp , and let g ∈ H+ . Then there is a T -structured subsequence Ah of Ag which is JhT -Fredholm.

Now, also the assertions of Theorems 2.31, 2.32 and 2.33 hold true for all sequences A ∈ FTlp having the additional property that one P-regularizer of W 0 (A) (if one exists) automatically belongs to Tlp .10 That means, we have criteria for the stability and Fredholm property of such sequences, and we know that the asymptotic behavior of the approximation numbers and indices of their entries is closely connected with the kernel dimensions and indices of the snapshots. Actually, the need for the additional condition on the regularizers B of W 0 (A) stems from the proof of Proposition 2.28, where we use that for both, W 0 (A) and B, the relation (4.14) holds. This can be achieved if we assume that W 0 (A) fulfills, besides (4.14), also the dual relation for kLn−m W 0 (A)(I − Ln )k: Definition 4.34. An operator A ∈ L(lp ) is called weakly band-dominated (w.r.t. (Ln )), if lim sup{k(I − Ln )ALn−m k, kLn−m A(I − Ln )k : n > m} = 0

m→∞

holds. The set of all weakly band-dominated operators shall be denoted by Wlp .

10 This even covers the above counterexamples for the “quasi-triangular setting”, but it remains as an open question if, or under which conditions the regularizers of a triangle-dominated operator are of that type again.

112


Since this is exactly what we know as P-centralized elements from Section 3.3.3, Wlp proves to be a Banach subalgebra of L(lp , P). For FWlp , the smallest closed algebra which contains all finite section sequences of rich weakly band-dominated operators, we again adapt Proposition 2.28 and take account of Proposition 3.31 to prove the following Proposition 4.35. Let A ∈ FWlp be such that W 0 (A) is P-Fredholm and let g ∈ H+ . Then there is a T -structured subsequence Ah of Ag which is JhT -Fredholm. This permits to replace Alp by Wlp in the Theorems 2.31, 2.32 and 2.33 without any further restrictions. We note that clearly Alp ⊂ Wlp by Theorem 1.55 and the inclusion is proper in general as the next section reveals.

4.4.7

Toeplitz operators with symbol in Cp + Hp∞

After this short course on generalizations of band-dominated operators on lp (Z, X) we again look at Toeplitz operators on lp (N, C), and we mention that the notions of triangle-dominated and weakly band-dominated operators translate to these spaces in a natural way, as in Section 4.4.1. Here, only the spaces with 1 < p < ∞ are of interest, since M 1 = C 1 = M ∞ = C ∞ = W are already completely resolved. Notice that the set of compact and the set of P-compact operators coincide for every 1 < p < ∞, respectively.

Let H ∞ stand for the set of all functions a ∈ L∞ (T) whose Fourier coefficients fulfill ai = 0 for all i > 0, and set Hp∞ := M p ∩ H ∞ .11 Furthermore, let χ1 denote the function χ1 (x) = x. From [12], 2.51 we know “It is obvious that Hp∞ is a closed subalgebra of M p . Let alg(C p , Hp∞ ) denote the smallest closed subalgebra of M p containing C p and Hp∞ . It is clear that alg(C p , Hp∞ ) coincides with alg(χ1 , Hp∞ ). The discontinuous function (1 − χ1 )iβ (β ∈ R \ {0}) can be shown to belong to Hp∞ for all 1 < p < ∞. ([12], Theorem 6.42 et seq.) This shows that alg(C p , Hp∞ ) is strictly larger than C p .” Moreover, Sarason [77] proved that this algebra even coincides with C p + Hp∞ := {f + g : f ∈ C p , g ∈ Hp∞ },

(4.15)

and we use this notation in what follows. Böttcher and Silbermann observed that for a function a ∈ C p + Hp∞ the Toeplitz operator T (a) is Fredholm on lp (N, C) if and only if a is invertible in C p + Hp∞ . In this case T (a−1 ) is a regularizer for T (a). (see [12], Theorems 2.53, 2.60). Also recall Coburns theorem ([12], Theorem 2.38) which yields that every Fredholm Toeplitz operator is one-sided invertible. We condense the previous observations to the following generalization of both Theorem 4.28 and the stability criterion in [12], Theorem 7.20. Theorem 4.36. Let A = {An } := {Ln ALn } with A = T (a) + K, a ∈ C p + Hp∞ and K compact. Then A is Fredholm if and only if A is Fredholm. In this case the approximation numbers of the An have the splitting numbers α(A) = β(A) = max{dim ker A, dim coker A}. Otherwise, all approximation numbers tend to zero as n → ∞. A is stable if and only if A is invertible. Proof. Clearly, A belongs to Tlp (N) , the set of all triangle-dominated operators on lp (N), and if A is Fredholm then the regularizer T (a−1 ) is in Tlp (N) , too. The snapshot JW 1 (A)J, as discussed ˜(t) := a(t−1 ). Hence, in Section 4.4.1, is the Toeplitz operator with the symbol a ˜ ∈ C p + Hp∞ , a the Fredholmness of T (a) and Proposition 4.33 imply the Fredholmness of W 1 (A), and Coburns theorem applied to JW 1 (A)J even provides its one-sided invertibility. The rest is as in the proof of Theorem 4.26. 11 Analogously,

H ∞ consists of all functions a with vanishing Fourier coefficients ai , i < 0.


113

This particularly covers the Toeplitz operators with so-called quasi continuous symbols, that are functions in QC p := (C p + Hp∞ ) ∩ (C p + Hp∞ ). Notice that such Toeplitz operators are weakly band-dominated and, by [12], 2.80, C 2 is a proper subset of QC 2 . For some further reading we also refer to [76], or Appendix 4 in [52], and the literature cited there. Remark 4.37. We also want to mention that there is an analogon to Sarasons observation (4.15) for band-dominated operators. We denote the set of all rich triangular operators in L(lp (N, C), P) by ∆lp (N) and note that it forms a Banach algebra. Now, it also holds that Alp (N) + ∆lp (N) = alg{Alp (N) , ∆lp (N) },

(4.16)

hence this sum is a Banach algebra as well. To prove this we proceed in the same way as in [12] for the Toeplitz case. We recall the mappings SR which were defined by (1.18) and whose compressions SR : L(lp (N), P) → L(lp (N), P) are well defined, linear and bounded by 2 as well. Proposition 1.56 further yields that SR (A) are band operators and kSR (A) − Ak → 0 as R → ∞ for every band-dominated A. Moreover, SR (A) are triangular whenever A is triangular, and if for h = (hn ) ∈ H+ the limit operator Ah exists then there is a subsequence g ⊂ h such that (SR (A))g exists as well (Consider the decompositions hn = cn R + rn with cn ∈ N and remainders 0 ≤ rn < R, and choose the subsequence g such that the respective remainders converge). If, in particular, A ∈ L(lp (N), P) is rich then all SR (A) are rich. Having these properties of the mappings SR ∈ L(L(lp (N), P)) we can apply a lemma of Zalcman and Rudin (see [12], 2.52) to prove that Alp (N) + ∆lp (N) is a closed linear space. If Ai = Bi + Ti , i = 1, 2 with Bi ∈ Alp (N) and Ti ∈ ∆lp (N) then the operators AR i := SR (Bi ) + Ti converge to R p Ai in the norm, respectively. Clearly, AR A belong to A + ∆ l (N) lp (N) and converge to A1 A2 as 1 2 R → ∞, hence A1 A2 ∈ Alp (N) + ∆lp (N) as well, that is (4.16) is proved.

4.4.8

A larger class of slowly oscillating operators

We finally enlarge the class of slowly oscillating operators by calling an operator A ∈ Tlp (N) slowly oscillating if it is rich and all its limit operators are shift invariant, that is Vk Ah V−k = Ah

for all k ∈ Z and all Ah ∈ σop (A).

Since for every sequence A := {Ln ALn } with a slowly oscillating operator A and every h ∈ HA the snapshot JW 1 (Ah )J is a Toeplitz operator, we can plug Coburns theorem (which provides onesided invertibility for every Fredholm Toeplitz operator) into the results for triangle-dominated operators. In analogy to Theorem 4.26 we then obtain Theorem 4.38. Let A be a slowly oscillating operator and A := {Ln ALn } be the sequence of its finite sections. If A is a Fredholm operator with a triangle-dominated regularizer then A is Fredholm. If A is invertible in Tlp (N) then A is stable. Conversely if A is Fredholm then A is Fredholm and α(A) = β(A) = max{dim ker A, dim coker A} are splitting numbers of A. It is not clear at the moment if the condition on the regularizer/the inverse of A can be dropped in general. At least, there is a positive answer if A = B + K where K is P-compact and B decomposes into a finite product of operators which are one of the following • a Fredholm weakly band-dominated operator,

• a Fredholm Toeplitz operator with symbol in C p + Hp∞ , • a sum I + C with a triangle-dominated C having norm less than 1, • an invertible quasi-triangular operator.

114


Index – Symbols – A∗ , adjoint operator. . . . . . . . . . . . . . . . . . . . . . . . .3 A? , Hilbert adjoint operator . . . . . . . . . . . . . . . . . 3 A/J , quotient algebra . . . . . . . . . . . . . . . . . . . . . . 2 ALp , band-dominated operators . . . . . . . . . . . . 87 Alp , band-dominated operators. . . . . . . . . . . . .34 α(A), α-number of A . . . . . . . . . . . . . . . . . . . 47, 51 Al,u n , adapted finite section . . . . . . . . . . . . . 58, 91 A(P), P-compatible elements . . . . . . . . . . . . . . 76 A(P, C), P-centralized elements . . . . . . . . . . . . 79 A(P, K), P-compact elements . . . . . . . . . . . . . . 76 B (S), neighborhood of S . . . . . . . . . . . . . . . . . . 27 β(A), β-number of A . . . . . . . . . . . . . . . . . . . 47, 51 Bm (A), Bernstein number . . . . . . . . . . . . . . . . . . 21 bγn , inflating spline . . . . . . . . . . . . . . . . . . . . . . 74, 88 BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 17, 81 Bp , Bp0 , convolution operators . . . . . . . . . . . . . . 92 BUC, bdd and uniformly cts functions . . . . . 35 C, complex numbers . . . . . . . . . . . . . . . . . . . . . . . . 2 C, localizing setting . . . . . . . . . . . . . . . . . . . . . . . . 64 C(a), convolution operator . . . . . . . . . . . . . 92, 94 cen A, center of A . . . . . . . . . . . . . . . . . . . . . . . . . . 63 coker A, cokernel of A . . . . . . . . . . . . . . . . . . . . . . . 3 Com(A), commutant of A . . . . . . . . . . . . . . . . . . 64 cond A, condition number of A . . . . . . . . . . . . . 69 CP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9, 77 C p + Hp∞ , algebra of functions . . . . . . . . . . . . 112 C(R2 ), continuous functions . . . . . . . . . . . . . . . . 94 dH (S, T ), Hausdorff distance . . . . . . . . . . . . . . . 32 δij , Kronecker delta . . . . . . . . . . . . . . . . . . . . 17, 23 dist(F, G), distance of sets . . . . . . . . . . . . . . . . . 35 DP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 F, algebra of sequences . . . . . . . . . . . . . . . . . . . . 40 FALp , finite section algebra of BdOs . . . . . . . 88 FAlp , finite section algebra of BdOs . . . . . . . . 55 F(A, P), P-convergent sequences. . . . . . . . . . .80 F(X, P), P-convergent operator sequences . 10 F T , algebra of sequences with snapshots . . . 40 Fx , local filter function . . . . . . . . . . . . . . . . . . . . . 64

G, ideal in F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Γm,n N, , sublevel set . . . . . . . . . . . . . . . . . . . . . . . . . . 27 m,n m γN , γN , γN , γ, level functions . . . . . . . . 27, 28 H± , strictly (in/de)creasing sequences . . . . . . 55 HA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 I, ideal of compact sequences . . . . . . . . . . . . . . 50 im A, range of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 ind A, index of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 ind A, index of the sequence A . . . . . . . . . . . . . 47 ind± A, plus-(minus-)index . . . . . . . . . . . . . . . . . 91 J, flip operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 j(A), injection modulus . . . . . . . . . . . . . . . . . . . . 21 J t , J T , ideals of sequences . . . . . . . . . . . . . . . . 41 JV , embedding map . . . . . . . . . . . . . . . . . . . . . . . . 21 Jx , local ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ker A, kernel of A . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 K(X), K(X, Y), compact operators . . . . . . . . . . 3 K(X, P), P-compact operators . . . . . . . . . . . . . . 9 Ll,u n , adapted finite section projection. . .58, 91 Lhn , finite section projection . . . . . . . . . . . . . . . 55 L$ (lp , P), rich operators. . . . . . . . . . . . . . . . . . . .38 lp = lp (Z, X), l2 , l∞ , sequence spaces . . . . . . . . 7 Lp = Lp (R, Y ), L2 , L∞ , function spaces . . . . 87 LT , localizable sequences . . . . . . . . . . . . . . . . . . . 64 L(X), L(X, Y), bounded linear operators . . . 2 L(X, P), P-compatible operators . . . . . . . . . . . . 9 MC , maximal ideal space . . . . . . . . . . . . . . . . . . 62 Mm (A), Mityagin number . . . . . . . . . . . . . . . . . . 21 M p , multiplier algebra . . . . . . . . . . . . . . . . . . . . 106 N, natural numbers, 1, 2, . . . . . . . . . . . . . . . . . . . 2 P , projection “onto lp (N)” . . . . . . . . . . . . . . . . .105 P, approximate projection . . . . . . . . . . . . . . . 8, 76 P-lim An , P-strong limit . . . . . . . . . . . . . . . 10, 80 p-lim Mn , partial limiting set . . . . . . . . . . . . . . . 70 Pm , projection on lp . . . . . . . . . . . . . . . . . . . . . . . . 7 ϕt (x), ϕt,r (x), inflated functions . . . . . . . . . . . 35 φx (A), local homomorphism . . . . . . . . . . . . . . . . 63 q(A), surjection modulus . . . . . . . . . . . . . . . . . . . 21 QC p , quasi continuous functions . . . . . . . . . . 113 QW , quotient map . . . . . . . . . . . . . . . . . . . . . . . . . 21

116

INDEX

R, real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 R± , non-negative (non-positive) reals . . . . . . . . 2 ρt , local diffeomorphism . . . . . . . . . . . . . . . . . . . . 94 rank A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Rnt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 RT , rich sequences . . . . . . . . . . . . . . . . . . . . . . . . . 53 σm (A), singular value . . . . . . . . . . . . . . . . . . . . . . 24 s-lim An , strong limit . . . . . . . . . . . . . . . . . . . . . . . 6 srm (A), slm (A), approximation numbers . . . . . 21 sp A, spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 sp A, spN, A, pseudospectra . . . . . . . . . . . . . . . 26 spess A, essential spectrum . . . . . . . . . . . . . . . . . 26 σop (A), operator spectrum . . . . . . . . . . . . . 37, 87 σstab (A), stability spectrum . . . . . . . . . . . . . . . . 57 T = {Tn }, truncation projections . . . . . . . . . . . 64 T (a), Toeplitz operator . . . . . . . . . . . . . . . . . . . 106 T RT , rich sequences . . . . . . . . . . . . . . . . . . . . . . . 73 T T , truncated localizable sequences . . . . . . . . 64 Uα , shift operator . . . . . . . . . . . . . . . . . . . . . . . . . . 87 u-lim Mn , uniform limiting set. . . . . . . . . . . . . .70 Vα , shift operator . . . . . . . . . . . . . . . . . . . . . . . . . . 34 VSO, very slowly oscillating functions . . . . . 103 Wn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 W (R2 ), Wiener algebra . . . . . . . . . . . . . . . . . . . . 94 W t (A), snapshot of A . . . . . . . . . . . . . . . . . . . . . . 40 X0 , subspace of X . . . . . . . . . . . . . . . . . . . . . . . . . 17 Z, integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Z± , non-negative (non-positive) integers . . . . . 2 –A– algebra Banach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 C ∗ -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Calkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 KMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 sequence algebras. . . . . . . . . . . . . . . . . . . . . .40 with approximate projection P . . . . . . . . 76 approximate projection . . . . . . . . . . . . . . . . . . 8, 76 approximate identity . . . . . . . . . . . . . . . 10, 80 equivalent . . . . . . . . . . . . . . . . . . . . . . . . . 10, 77 uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9, 77 approximation numbers . . . . . . . . . . . . . . . . . . . . 21

band-dominated operator . . . . . . . . . . . . . . . 34, 87 Bernstein numbers . . . . . . . . . . . . . . . . . . . . . . . . . 21 –C– C ∗ -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Calkin algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 cokernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 commutant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 complete w.r.t. P-strong convergence . . . . . . 12 complex uniformly convex space . . . . . . . . . . . . 70 condition numbers . . . . . . . . . . . . . . . . . . . . . . . . . 69 cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 convergence ∗-strong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 P-strong . . . . . . . . . . . . . . . . . . . . . . . 10, 12, 80 strong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 –D– dual space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 –F– Fredholm operator . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Fredholm sequence . . . . . . . . . . . . . . . . . . . . . . . . . 51 functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 –G– Gelfand transform . . . . . . . . . . . . . . . . . . . . . . . . . 62 Gelfand-Naimark theorem . . . . . . . . . . . . . . . . . . 62 –H– Hausdorff distance . . . . . . . . . . . . . . . . . . . . . . . . . 32 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 –I–

ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 maximal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62 index operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 plus-(minus-)index . . . . . . . . . . . . . . . . . . . . 91 sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 injection modulus . . . . . . . . . . . . . . . . . . . . . . . . . . 21 inverse closedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 –B– invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Banach algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Banach space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 invertible at infinity . . . . . . . . . . . . . . . . . . . . . . . . . 9 Banach-Stone theorem . . . . . . . . . . . . . . . . . . . . . 62 involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

INDEX

117

P-centralized elements . . . . . . . . . . . . . . . . . . . . . 79 P-compactness . . . . . . . . . . . . . . . . . . . . . . . . . . 9, 76 kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 P-dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 KMS-algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 P-Fredholmness . . . . . . . . . . . . . . . . . . . . . . . . . 9, 79 P-regularizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9, 76 projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 –L– pseudospectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 limit operator . . . . . . . . . . . . . . . . . . . . . . . . . . 37, 87 P-strong convergence . . . . . . . . . . . . . . . . . . . 10, 80 localizable sequence . . . . . . . . . . . . . . . . . . . . . . . . 64 –Q– localizing setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 faithful . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 quasi continuous functions . . . . . . . . . . . . . . . . 113 locally, globally . . . . . . . . . . . . . . . . . . . . . . . . 64 quotient algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 –K–

–M–

–R–

maximal ideal space . . . . . . . . . . . . . . . . . . . . . . . . 62 range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Mityagin numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 21 rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 rich operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 37, 87 –O– –S– Ω-periodic functions . . . . . . . . . . . . . . . . . . . . . . 103 operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 sequence adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 α-, β-number . . . . . . . . . . . . . . . . . . . . . . 47, 51 band-dominated . . . . . . . . . . . . . . . . . . . 34, 87 almost uniformly bounded rank . . . . . . . . 50 compact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 C-localizable . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 convolution . . . . . . . . . . . . . . . . . . . . . . . . 92, 94 compact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Fredholm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 cross-dominated . . . . . . . . . . . . . . . . . . . . . . 108 index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Fredholm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 J T -Fredholm. . . . . . . . . . . . . . . . . . . . . . . . . .42 Hilbert adjoint . . . . . . . . . . . . . . . . . . . . . . . . . 3 nullity, deficiency . . . . . . . . . . . . . . . . . . . . . . 47 invertible at infinity . . . . . . . . . . . . . . . . . . . . 9 regular J T -Fredholm . . . . . . . . . . . . . . . . . . 47 locally compact. . . . . . . . . . . . . . . . . . . . . . . .91 splitting property. . . . . . . . . . . . . . . . . . . . . .49 normally solvable . . . . . . . . . . . . . . . . . . . . . . . 3 stable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7, 50 of multiplication . . . . . . . . . . . . . . . . . . . . . . . 34 T -rich, rich . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 P-compact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 T -structured . . . . . . . . . . . . . . . . . . . . . . . . . . 53 P-Fredholm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 singular values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 properly P-deficient . . . . . . . . . . . . . . . . . . . 14 spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 essential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 properly P-Fredholm . . . . . . . . . . . . . . . . . . 14 pseudospectrum . . . . . . . . . . . . . . . . . . . . . . . 26 quasi-diagonal . . . . . . . . . . . . . . . . . . . . . 33, 84 quasi-triangular . . . . . . . . . . . . . . . . . . . . . . 109 splitting property . . . . . . . . . . . . . . . . . . . . . . . . . . 49 rich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37, 87 stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7, 50 self-adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 surjection modulus . . . . . . . . . . . . . . . . . . . . . . . . . 21 shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 –T– slowly oscillating . . . . . . . . . . . . . . . . . . . . . 113 Toeplitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Toeplitz matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 triangle-dominated . . . . . . . . . . . . . . . . . . . 111 –U– weakly band-dominated . . . . . . . . . . . . . . 111 operator spectrum . . . . . . . . . . . . . . . . . . . . . . 37, 87 uniform limiting set . . . . . . . . . . . . . . . . . . . . . . . . 70 –P–

–V–

partial limiting set . . . . . . . . . . . . . . . . . . . . . . . . . 70

very slowly oscillating functions . . . . . . . . . . . 103

118

INDEX

Bibliography [1] G. R. Allan, Ideals of vector-valued functions, Proc. London Math. Soc. 18 (1968), 193-216. [2] G. R. Allan, Ten lectures on operator algebras, CBMS Reg. Conf. Ser. Math., 55, Amer. Math. Soc., Providence, RI, 1984. [3] I. D. Berg, An extension of the Weyl-von Neumann theorem to normal operators, Trans. Amer. Math. Soc. 160 (1971), 365-371. [4] A. Böttcher, Norms of inverses, spectra, and pseudospectra of large truncated Wiener-Hopf operators and Toeplitz matrices, New York J. Math. 3 (1997), 1-31. [5] A. Böttcher, On the approximation numbers of large Toeplitz operators, Documenta Mathematica 2 (1997), 1-29. [6] A. Böttcher, Pseudospectra and singular values of large convolution operators, J. Integral Equations Appl. 6 (1994), No. 3, 267-301. [7] A. Böttcher, Two dimensional convolutions in corners with kernels having support in a halfplane, Mat. Zametki 34 (1983), No. 2, 207-218; Math. Notes 34 (1983), 585-591. [8] A. Böttcher, S. M. Grudsky, Spectral properties of banded Toeplitz matrices, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. [9] A. Böttcher, S. M. Grudsky, B. Silbermann, Norms of inverses, spectra, and pseudospectra of large truncated Wiener-Hopf operators and Toeplitz matrices, New York J. Math. 3 (1997), 1-31. [10] A. Böttcher, N. Krupnik, B. Silbermann, A general look at local principles with special emphasis on the norm computation aspect, Integral Equations Oper. Theory 11 (1988), No. 4, 455-479. [11] A. Böttcher, M. Lindner, Pseudospectrum, Scholarpedia, 3(3):2680, 2008. [12] A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, 2nd Edition, Springer-Verlag, Berlin, 2006. [13] A. Böttcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, SpringerVerlag, New York, 1999. [14] N. P. Brown, Quasi-diagonality and the finite section method, Math. Comp. 76 (2007), No. 257, 339-360.

120

BIBLIOGRAPHY

[15] S. N. Chandler-Wilde, M. Lindner, Limit Operators, Collective Compactness and the Spectral Theory of Infinite Matrices, Mem. Amer. Math. Soc. 210 (2011), No. 989. [16] S. N. Chandler-Wilde, M. Lindner, Boundary integral equations on unbounded rough surfaces: Fredholmness and the Finite Section Method, J. Integral Equations Appl. 20 (2008), 13-48. [17] J. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396-414. [18] A. Daniluk, The maximum principle for holomorphic operator functions, Integral Equations Oper. Theory 69 (2011), No. 3, 365-372. [19] R. G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, London 1972. [20] R. G. Douglas, R. Howe, On the C ∗ -algebra of Toeplitz operators on the quarter plane, Trans. Amer. Math. Soc. 158 (1971) 203-217. [21] N. Dunford, J. T. Schwartz, Linear Operators. Part I: General Theory, Interscience Publishers, Inc., New York, Interscience Publishers, Ltd., London 1958; John Wiley & Sons Ltd., New York, 1988. [22] D. E. Edmunds, W. D. Evans, Spectral Theory and Differential Operators, Oxford University Press, New York, 1987. [23] L. S. Goldenstein, I. C. Gohberg, On a multidimensional integral equation on a half-space whose kernel is a function of the difference of the arguments, and on a discrete analogue of this equation, Dokl. Akad. Nauk SSSR 131 (1960), 9-12; Sov. Math. Dokl. 1 (1960), 173-176. [24] I. C. Gohberg, I. A. Feldman, Convolution equations and projection methods for their solution Nauka, Moscow 1971; Engl. transl.: Amer. Math. Soc. Transl. of Math. Monographs 41, Providence, R. I., 1974; German transl.: Akademie-Verlag, Berlin 1974. [25] I. C. Gohberg, N. Krupnik, One-dimensional Linear Singular Integral Equations, Birkhäuser Verlag, Basel, Boston, Berlin, 1992. [26] J. Globevnik, Norm-constant analytic functions and equivalent norms, Illinois J. Math. 20 (1976), 503-506. [27] J. Globevnik, On complex strict and uniform convexity, Proc. Amer. Math. Soc. 47 (1975), 176-178. [28] P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887-933. [29] R. Hagen, S. Roch, B. Silbermann, Spectral Theory of Approximation Methods for Convolution Equations, Operator Theory: Advances and Appl. 74, Birkhäuser Verlag, Basel, Boston, Berlin 1995. [30] R. Hagen, S. Roch, B. Silbermann, C ∗ -Algebras and Numerical Analysis, Marcel Dekker, Inc., New York, Basel, 2001. [31] A. C. Hansen, Infinite-dimensional numerical linear algebra: theory and applications, Proc. R. Soc. Lond. Ser. A 466, No. 2124 (2010), 3539-3559. [32] A. C. Hansen, On the approximation of spectra of linear operators on Hilbert spaces, J. Funct. Anal. 254 (2008), No. 8, 2092-2126.

BIBLIOGRAPHY

121

[33] A. C. Hansen, On the solvability complexity index, the n-pseudospectrum and approximations of spectra of operators, J. Amer. Math. Soc. 24 (2011), 81-124. [34] A. C. Hansen, O. Nevanlinna, Complexity Issues in Computing Spectra, Pseudospectra and Resolvents, http://www.damtp.cam.ac.uk/user/na/people/Anders/Banach_Hansen_ Nevanlinna.pdf, Banach Center Publ., (accepted). [35] F. Hausdorff, Set Theory, Chelsea, New York, 1957. [36] G. Heinig, F. Hellinger, The finite section method for Moore-Penrose inversion of Toeplitz operators, Integral Equations Oper. Theory 19 (1994), No. 4, 419-446. [37] P. Junghanns, S. Roch, B. Silbermann, Collocation methods for systems of Cauchy singular integral equations on an interval, Comp. Techn. 6 (2001), 88-124. [38] P. Junghanns, G. Mastroianni, M. Seidel, On the stability of collocation methods for Cauchy singular integral equations in weighted Lp -spaces, Math. Nachr. 283 (2010), No. 1, 58-84. [39] A. V. Kozak, A local principle in the theory of projection methods, Dokl. Akad. Nauk SSSR 212 (1973), 6, 1287-1289; Engl. transl.: Sov. Math. Dokl. 14 (1974), 5, 1580-1583. [40] H. Landau, On Szegö’s eigenvalue distribution theorem and non-Hermitian kernels, J. Anal. Math. 28 (1975), 335-357. [41] H. Landau, The notion of approximate eigenvalues applied to an integral equation of laser theory, Q. Appl. Math. (April 1977), 165-171. [42] B. V. Lange, V. S. Rabinovich, On the Noether property of multidimensional discrete convolutions, Mat. Zametki 37 (1985), No. 3, 407-421. [43] M. Lindner, The finite section method and stable subsequences, Appl. Num. Math. 60 (2010), No. 4, 501-512. [44] M. Lindner, Infinite Matrices and their Finite Sections, Birkhäuser Verlag, Basel, Boston, Berlin, 2006. [45] M. Lindner, Limit Operators and Applications on the Space of Essentially Bounded Functions, Dissertationsschrift, Chemnitz, 2003. [46] M. Lindner, V. S. Rabinovich, S. Roch, Finite sections of band operators with slowly oscillating coefficients, Lin. Alg. Appl. 390 (2004), 19-26. [47] V. A. Malysev, Wiener-Hopf equations in a quadrant of the plane, discrete groups and automorphic functions, Mat. Sb. 84 (1971), No. 126, 499-525; English transl.: Math. USSR Sb. 13 (1971). [48] H. Mascarenhas, Convolution type operators on cones and asymptotic spectral theory, Dissertationsschrift, Chemnitz, 2004. [49] H. Mascarenhas, B. Silbermann, Convolution type operators on cones and their finite sections, Math. Nachr. 278 (2005), No. 3, 290-311. [50] H. Mascarenhas, B. Silbermann, Spectral approximation and index for convolution type operators on cones on Lp (R2 ), Integral Equations Oper. Theory 65 (2009), No. 3, 415-448.

122

BIBLIOGRAPHY

[51] E. A. Maximenko, Convolution operators on expanding polyhedra: the limits of norms of inverse operators and pseudospectra, Mat. Sb. 44 (2003), No. 6, 1027-1038. [52] N. K. Nikolski, Treatise on the shift operator. Spectral function theory. With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller, Translated from the Russian by Jaak Peetre, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 273, Springer-Verlag, Berlin, 1986. [53] A. Pietsch, History of Banach spaces and linear operators, Birkhäuser Boston, Inc., Boston, MA, 2007. [54] A. Pietsch, Operator Ideals, Verlag der Wissenschaften, Berlin, 1978. [55] N. I. Polski, Projection methods for solving linear equations, Uspekhi Mat. Nauk 18 (1963), 179-180 (Russian). [56] S. Prössdorf, B. Silbermann, Numerical Analysis for Integral and Related Operator Equations, Birkhäuser Verlag, Basel, Boston, Berlin, 1991. [57] V. S. Rabinovich, S. Roch, Algebras of approximation sequences: Spectral and pseudospectral approximation of band-dominated operators, Preprint 2186, Technical University Darmstadt, 2001. [58] V. S. Rabinovich, S. Roch, The Fredholm index of locally compact band-dominated operators on Lp (R), Integral Equations Oper. Theory 57 (2007), 263-281. [59] V. S. Rabinovich, S. Roch, J. Roe, Fredholm indices of band-dominated operators, Integral Equations Oper. Theory 49 (2004), No. 2, 221-238. [60] V. S. Rabinovich, S. Roch, B. Silbermann, Algebras of approximation sequences: Finite sections of band-dominated operators, Acta Appl. Math. 65 (2001), No. 3, 315-332. [61] V. S. Rabinovich, S. Roch, B. Silbermann, Band-dominated operators with operator-valued coefficients, their Fredholm properties and finite sections, Integral Equations Oper. Theory 40 (2001), No. 3, 342-381. [62] V. S. Rabinovich, S. Roch, B. Silbermann, Fredholm theory and finite section method for band-dominated operators, Integral Equations Oper. Theory 30 (1998), No. 4, 452-495. [63] V. S. Rabinovich, S. Roch, B. Silbermann, Limit Operators and Their Applications in Operator Theory, Birkhäuser Verlag, Basel, Boston, Berlin, 2004. [64] V. S. Rabinovich, S. Roch, B. Silbermann, On finite sections of band-dominated operators, Operator Theory: Advances and Applications 181 (2008), 385-391. [65] V. Rakočević, J. Zemánek, Lower s-numbers and their asymptotic behaviour, Studia Math. 91 (1988), No. 3, 231-239. [66] M. Reed, B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York, 1978. [67] L. Reichel, L. N. Trefethen, Eigenvalues and pseudo-eigenvalues of Toeplitz matrices, Lin. Alg. Appl. 162 (1992), 153-185. [68] S. Roch, Band-dominated operators on lp -spaces: Fredholm indices and finite sections, Acta Sci. Math. (Szeged) 70 (2004), 3/4, 783-797.

BIBLIOGRAPHY

123

[69] S. Roch, Finite Sections of Band-Dominated Operators, Memoirs of the AMS, 191 (2008), No. 895. [70] S. Roch, Algebras of approximation sequences: Fredholmness, J. Oper. Theory 48 (2002), 121-149. [71] S. Roch, P. Santos, B. Silbermann, Finite section method in some algebras of multiplication and convolution operators and a flip, Z. Anal. Anwendungen 16 (1997), No. 3, 575-606. [72] S. Roch, P. Santos, B. Silbermann, Non-commutative Gelfand Theories, Universitext, Springer-Verlag London, 2011. [73] S. Roch, B. Silbermann, Non-strongly converging approximation methods, Demonstr. Math. 22 (1989), 3, 651-676. [74] A. Rogozhin, B. Silbermann, Approximation numbers for the finite sections of Toeplitz operators with piecewise continuous symbols, J. Funct. Anal. 237 (2006), No. 1, 135-149. [75] A. Rogozhin, B. Silbermann, Banach algebras of operator sequences: Approximation numbers, J. Oper. Theory 57 (2007), No. 2, 325-346. [76] D. Sarason, Algebras of functions on the unit circle, Bull. Amer. Math. Soc. 79 (1973), 286-299. [77] D. Sarason, Generalized interpolation in H ∞ , Trans. Amer. Math. Soc. 127 (1967), 179-203. [78] M. Seidel, On (N, )-pseudospectra and quasi-diagonal operators, Preprint series TUChemnitz, Preprint 2011-13, 2011. [79] M. Seidel, B. Silbermann, Finite sections of band-dominated operators: lp -theory, Complex Analysis and Operator Theory 2 (2008), 683-699. [80] M. Seidel, B. Silbermann, Banach algebras of structured matrix sequences, Lin. Alg. Appl. 430 (2009), 1243-1281. [81] M. Seidel, B. Silbermann, Banach algebras of operator sequences, (accepted for publication in Operators and Matrices). [82] M. Seidel, B. Silbermann, Finite sections of band-dominated operators - Norms, condition numbers and pseudospectra, (accepted for publication in Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel). [83] B. Silbermann, Lokale Theorie des Reduktionsverfahrens für Toeplitzoperatoren, Math. Nachr. 104 (1981), No. 1, 137-146. [84] E. Shargorodsky, On the level sets of the resolvent norm of a linear operator, Bull. London Math. Soc. 40 (2008), 493-504. [85] I. B. Simonenko, Convolution type operators in cones, Mat. Sb. 74 (1967), No. 116, 298-313. [86] I. B. Simonenko, On multidimensional discrete convolutions, Mat. Issled. 3 (1968), 1, 108127 (Russian). [87] L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, Princeton, NJ, 2005.

124

BIBLIOGRAPHY

Theses on the thesis “On some Banach Algebra Tools in Operator Theory” presented by Dipl.-Math. Markus Seidel (1) This text is concerned with the investigation of operator sequences A = {An } which typically arise from approximation methods for bounded linear operators. The heart of the present approach is to capture the asymptotic properties of such sequences in terms of so-called snapshots, a family of operators where each of them displays one basic facet of the asymptotics, but together they give a comprehensive picture of A. The snapshots are usually taken by a slight transformation of the sequence and a limiting process. One may interpret this as looking at A from different angles, and one is interested in results as • The stability of a sequence is equivalent to the invertibility of all snapshots. • A sequence {An } is almost stable (we say Fredholm) iff all snapshots are Fredholm. In this case the dimensions of the kernels and cokernels of the snapshots as well as their indices provide information on the splitting property and the indices of the An . • The norms kAn k converge to the maximum of the norms of the respective snapshots. Analogous conclusions hold for the norms of the inverses and the condition numbers. • The pseudospectra of the elements An approximate the union of the pseudospectra of all snapshots. (2) The stability plays a crucial role for the applicability of approximation methods which replace a linear equation Ax = b by (hopefully simpler) equations An xn = bn since, as a rule, it holds that the convergence of bn to b and the convergence of An to A together with the stability of {An } imply the existence of the solutions xn and their convergence to the solution x of the initial equation. Different types of convergence can be studied. (3) Besides the singular values which are well known and available for operators on Hilbert spaces, there are further geometric characteristics being their natural generalizations to the Banach space case, such as the so-called approximation numbers and the related Bernstein and Mityagin numbers. These characteristics provide the right language to describe the behavior of almost stable (i.e. Fredholm) sequences and constitute the connecting link between the Fredholm properties of the snapshots of an operator sequence {An } and the Fredholm properties of its entries An . More precisely, the so-called splitting property determines how many approximation numbers (or singular values in case of Hilbert spaces) of the An tend to zero as n gets large. Moreover, the indices of the An converge and the limit is nothing but the sum of the indices of all snapshots.

(4) The above observations particularly apply to band-dominated operators on sequence spaces lp or function spaces Lp with 1 ≤ p ≤ ∞, to quasi-diagonal operators and to convolution type operators on cones. They recover lots of well known results for more concrete classes of operators, such as Toeplitz operators, but translate also to larger and more general classes: quasi-triangular, triangle-dominated, or weakly band-dominated operators. (5) In that business it turns out to be extremely fruitful to replace the classical triple (compactness, Fredholmness, strong convergence) by a similar concept which defines the substitutes P-compactness, P-Fredholmness and P-strong convergence for operators on a Banach space X with the help of so-called uniform approximate identities P. One restricts considerations to the set L(X, P) of operators A which are compatible with P-compactness in the sense that the product of A with any P-compact operator is P-compact again. In this set P-Fredholmness means invertibility up to P-compact operators, and a sequence converges P-strongly iff, multiplied with a P-compact operator, this always turns into norm convergence. Often, the approximate identities are intrinsically given by an approximation or discretization method, e.g. a sequence of nested projections, and one may say that the P-notions are induced by the underlying method. (6) Actually, the set L(X, P) forms a beautiful and perfectly self-contained universe: It is a closed and inverse closed algebra, all P-regularizers (the “almost inverses”) of P-Fredholm operators are still included, and the P-strong limits of sequences in L(X, P) belong to that set again. The notion of invertibility at infinity, an alternative way how to generalize the classical Fredholm property, proves to coincide with P-Fredholmness. In fact, many concrete settings, such as lp or Lp spaces, provide the mentioned approximate identities in a very natural way (e.g. sequences of canonical projections), where the classical Fredholm property can be completely embraced by the P-notions, and large classes of operators (Toeplitz and Hankel operators, convolution type and even singular integral operators, band-dominated and quasi-diagonal operators) perfectly fit into this framework. (7) This alternative P-concept covers and generalizes the classical one in large parts, and it mimics most of the important interactions between approximation methods and the three basic notions compactness, Fredholmness and strong convergence, which are usually exploited. Nevertheless, it displays its full power in settings where the classical approach fails, e.g. if the discretizations An are of infinite rank, hence not compact in general, or in cases where the strong convergence is not available anymore (such as l∞ ). (8) The above mentioned Fredholm theory for sequences with a certain asymptotic structure (namely the existence of snapshots) can be embedded into a more general approach which does not require any structure and provides a transcendent notion of Fredholmness. Surprisingly, the approximation numbers still provide a characterizing description for this general Fredholm property via so-called α- and β-numbers, an analogon to the splitting property for the structured case. (9) Between these two models another one can be established. The so-called rich sequences do not need to be structured in the previous sense, but shall have a rich amount of structured subsequences, and hence they strike a balance and benefit from both models. This dramatically enlarges the class of sequences whose asymptotics can be captured by snapshots (or snapshots of subsequences, in this case) but keeps large parts of the desired results.

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