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BANACH BUNDLES AND LINEAR OPERATORS

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1975 Russ. Math. Surv. 30 115 (http://iopscience.iop.org/0036-0279/30/5/R03) View the table of contents for this issue, or go to the journal homepage for more

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Russian Math. Surveys 30:5 (1975), 115-175 From Uspekhi Mat. Nauk 30:5 (1975), 101-157

BANACH BUNDLES AND LINEAR OPERATORS M. G. Zaidenberg, S. G. Krein, P. A. Kuchment, and A. A. Pankov

This paper has the character of a survey. In it we consider questions in the theory of Fredholm (Noether) and semi-Fredholm operators on a Banach space that depend on a many-dimensional parameter. The paper divides naturally into two parts: the theory of operators depending continuously on a parameter (§ §1-2), and the theory of operators depending holomorphically on a parameter (§ §3-5). The theory of Banach bundles provides a natural apparatus for the study of these questions. In this connection we give a detailed account of a number of parts of this theory. In addition, we quote, with proofs, some fundamental facts in functional analysis that have been obtained in the past ten or twenty years and are not well represented in the monographs and text-books.

Contents

Introduction §1. Banach bundles §2. Operators that depend continuously on a parameter §3. Holomorphic Banach bundles §4. Operators that depend holomorphically on a parameter. Global questions §5. Operators that depend holomorphically on a parameter. Local and semi-local questions Notes on the literature and supplements References

115 117 125 139 151 159 166 169

Introduction

The present paper arises from results of the work of S. G. Krein's seminar on the theory of linear operators that depend on a parameter. In the study of such operators families of Banach spaces depending on a parameter (kernels, images, and so forth), which under certain conditions are Banach bundles, arise naturally. We consider in the paper the cases of continuous and holomorphic dependence on the parameter, although the majority of the results on the continuous case can be carried over easily to the case of smooth dependence. 115

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We also present certain fundamental facts in functional analysis (results of Bartle, Graves, Bishop, Allan, and Bungart) which are not well represented in the monographs and text-books. §1 contains the theory of Banach vector bundles and their finitedimensional subbundles. Here we give a proof of a theorem of Bartle and Graves on the continuous non-linear embedding of a factor-space in the original Banach space. In §2 we present the Atiyah-Janich index theory for compact families of Fredholm operators, as extended by Neubauer to Banach spaces. It should be noted that a number of important facts that are needed in constructing this theory are already contained in a notable paper of A. S. Shvarts. We also study the index of families of semi-Fredholm operators. Questions on the existence of one-sided and two-sided inverses, and of regularizers, depending continuously on the parameter, are also considered. We pass to the holomorphic case in §3. We first prove a fundamental result of Bishop on the structure of holomorphic functions with values in a Banach space. As corollaries, we obtain theorems on lifting abstract holomorphic functions and on the triviality of the cohomology of Stein manifolds with coefficients in the sheaf of germs of such functions. Further, we give an account of the elements of the theory of Banach analytic manifolds (basically following Douady). In the last subsection we quote important properties of holomorphic Banach bundles. The exposition is based on classical results of Grauert, extended to the Banach case by Bungart. Unfortunately, these results are presented without proofs (we would probably need a separate paper for them). In §4 we study operators that depend holomorphically on a parameter, also on the basis of results of Grauert and Bungart. One of the important results is a theorem of Allan on the existence of a holomorphic one-sided inverse (the plan of the proof is due to Shubin). Then we consider the question of the existence of holomorphic regularizers. We investigate the analytic structure of certain regions and subsets of spaces of operators (pseudoconvexity, analyticity). In the last subsection we prove theorems on the behaviour of global inverses and regularizers near a singularity. §5 is devoted to the study of the spectrum, cospectrum, and root functions of operators that depend holomorphically on a parameter. Here we consider the many-dimensional case only, and we do not touch on the large group of papers on operators that depend on a single parameter. We establish a theorem on the decomposition of a Fredholm operator-function into factors corresponding to the irreducible branches of its spectrum or cospectrum. Mastery of the elements of K-theory, up to Chapter 2 of [ 1 ] , and acquaintance with the concept of a locally-trivial bundle, are prerequisite

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to reading § § 1 —2. The elements of the theory of several complex variables, as contained in Chapters 1 and 3—5 of the second part of [60], are prerequisite for §3-5. We also assume a knowledge of the fundamentals of operator theory. All the basic references and some supplements are collected at the end of the paper, in the section "Notes on the literature and supplements". The authors thank all the participants in the seminar, particularly V. I. Ovchinnikov, V. P. Trofimov and I. Ya. Shneiberg, for valuable discussions. We also thank A. S. Shvarts for a number of useful critical remarks. § 1 . Banach bundles

0. The Bartle-Graves theorem on lifting continuous functions. Let Ε be a Banach space, Ν a closed subspace, and π: Ε -*• E/N the canonical projection of Ε onto the factor-space E/N. THEOREM 1.1 (Bartle and Graves [72]). There is a continuous homogeneous map p: E/N -*• Ε such that πρ(χ) = χ for every χ G E/N. Further, for each λ > 1 we can choose ρ so that \\ p(x) \\E < λ || χ \\Ε/Ν. PROOF. We need only construct a continuous map φ of the unit sphere S(E/N) into Ε such that ιτφ(χ) = χ and || φ(χ) \\Ε < λ (χ G S(E/N)). For then the map h(x) = || χ || φ Ι 1 (h(0) = 0) has all the required properties, MU 11/ except that it is only positively homogeneous. A homogeneous map ρ can now be defined by p(x) = j h(x) - -j Λ(— χ) in the real case, and by 2π

p(x) = j ^ \ e~"ph(e''px) άφ in the complex case. ο We write π " 1 ( ^ ) λ (x "€= S(E/N)) for the intersection of the preimage -1 π ( ^ ) with the ball {χ: χ e Ε, \\ χ || < λ}. This set is closed and convex. A continuous map φ: S(E/N) -*• Ε with the required properties can be defined if and only if ψ(χ) G π " 1 ^ for all χ G S(E/N). We construct this map as the limit of a sequence of maps φί for which \!/,(x) e U2-i (κ~χ(χ\)- (Here ί/ε(?Ι) denotes the ε-neighbourhood of a setSl). We start by constructing φλ. For x 0 G S(E/N) we choose an element x 0 G π " 1 ( χ 0 ) λ and consider the map φ1% (χ) = x 0 . On the sphere S(E/N) there is a neighbourhood V% of x 0 such that for all points χ of it we have Ψ 1 ά ο (χ) = x 0 G U ~I{K~ {X\). From the cover of the sphere by the neighbourhoods F 5 we select a locally-finite subcover V% , and a corresponding partition of unity ι?α(χ). Since U2-1(ir~l(x\) is convex, the map Φι(χ) = Σ ηα(χ) φ1% (χ) is as required. 1

2

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We now construct inductively a sequence of maps ψ,·(χ) such that ψ,.(χ) G U2-i(n-l(x\) and || φί+1(χ) - ψ;·(χ) || < 1/2'-1. By the inductive hypothesis the set U2-i(\jji(x)) Π π - 1 ( χ ) λ is non-empty and convex for each χ G S(E/N). We need only ensure that %+i(i) 6 U2-i-i (ί/ 2 -ί(Ψί(ί))ηπ- 1 (^) λ ). We can achieve this in a neighbourhood of each point x0 by means of a constant map Ψ,·+1 ; * ο ; we then "paste together" by means of a partition of unity, as was done above. It is clear that the sequence ψ,- converges. Let SC be a topological space. We write C(S,E) for the space of all bounded continuous functions / on SC with values in the Banach space E. This is a Banach space under the norm | | / | | c = sup ||/(τ) ||. The map

π: Ε -> E/N induces a map π,: C(S,E)

-+ C(SC,E/N).

The following theorem is an immediate consequence of Theorem 1.1. THEOREM 1.2 (on lifting). The map π* is an epimorphism of C(SC,E) onto C(SC,E/N). COROLLARY. Let SC be a CW-complex, 2/ a subcomplex and f. SC -> E/N and / i : 3/ -»· Ε continuous maps such that π ° f\ =J\oy.Then there is a continuous map f: SC -*• Ε such that f \y = fx and it ο f = f. To prove this we construct in accordance with Theorem 1.2 a map / 2 : 3C -> Ε such that π ο f2 = /· We can then consider the map (Λ — /2)12/; ^ -* ^ · Since Ν is contractible, we can extend this map to a map / 3 : SC -*• Ν (extension by skeleton). Then / = f2 + / 3 satisfies all the requirements of the corollary. 1. Banach vector bundles. Let SC and I be topological spaces and let p:%-+SC\>*i a continuous surjection (projection). We assume that the fibre ρ'ι{τ) = % over each point r £ S " i s given a Banach space structure for which the strong topology coincides with the topology induced from %. An open cover {Ua} of JTis called trivializing for ρ if for each a there is l a Banach space Ea and a homeomorphism φα: p~ (Ua) -* ί/α Χ £ α having the properties: 1) the diagram Φα

is commutative, where ν is the natural projection on Ua, and for each τ £ Ua the map φα induces on the fibre % T an isomorphism φατ of the Banach spaces MT and £ a . 2) If Ua and £/ff are two elements of the cover, then the map of Ua Π U0 to %(Ea, Εβ) given by τ -* ( ^ ° ^ Μ τ i s continuous in the operator norm. The maps φα are called trivializing and the maps 1 τ t n e τ -*· ( ^ ° Ψα Χ ~ Ψαβ( ) transition functions. Two trivializing covers are said to be equivalent if their union satisfies

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1) and 2). A class of equivalent trivializing covers gives the structure of a Banach vector bundle on (2£, %·, p). Here % is the bundle space, and .2^its base. If Sis connected, then all the Ea are necessarily isomorphic; that is, they can be identified with a particular Banach space E. In this case we say that % is a Banach vector bundle with fibre E. By a morphism A: %i -*• %2 of Banach vector bundles Pi

© ,· -» 3C (i = 1, 2) we understand a continuous map such that p2A = pv that at each point r £ 3C the induced map AT: % lT -*• $ 2 τ is linear and continuous, and that there are from § t to %2 s u c n that p2A = p l 5 that there are trivializing maps φ{ί): ρ,"1 (U) -»· ί/ Χ Ε' (i = 1, 2) (τ e {/) for which the map from i/ into X(EX, E2) given by τ -»· φ^ o Ατ ο (ψ^)~ι is continuous in the operator norm. An isomorphism of I , onto g 2 is a morphism which is a one-to-one Pi

map. In this case we say that ©Ί -* S" and g

P2 2

->· «^ are isomorphic.

ρ

A bundle isomorphic to the bundle JT X £ -»· X ( where ρ is the projection onto the first component) is said to be trivial. An important example of a trivial bundle is generated by the map Ε -*• E/N, where £ is a Banach space and Ν a closed subspace of it. To describe this bundle we use the map ρ of Theorem 1.1 to construct a map φ: Ε -*• Ε/Ν Χ Ν by the formula φ(χ) = (π(χ), χ - ροττ(χ)). This is an isomorphism. U = E/N is a trivializing cover and the map φ defines on E{N, E, n) the structure of a trivial Banach vector bundle, with fibre N. A section of a bundle % -> JT is a continuous map δ: SC -> % which is a right inverse of ρ (that is, ρ ο δ = /c?^- ). Let % -> J^ be a Banach vector bundle with fibre £ and let ^F be a subspace of g such that for each τ Ε ZV the intersection £ τ Π J^ is a closed linear subspace of e5,.. We write g for the restriction of ρ to J^and assume that a Banach vector bundle structure with fibre F is introduced on (S£, jf, q). This bundle is said to be a subbundle of % if for each point τ Ε 2C there are trivializations φ: ρ'1 (U) ->• U Χ Ε and ψ: q'1 (U) -+ U X F (τ e U) such that the map τ -> φτ ο (ψ,.)" 1 is constant on C/. A subbundle JF" is (i/T-eci if its fibres are complemented subspaces of the corresponding fibres of % . The following construction provides an important example of a direct subbundle. Suppose that on a Banach space Ε we are given a family of infinite-dimensional bounded projections Ρ(τ), depending norm-continuously on the point τ of a topological space 3C. We consider the trivial bundle

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% = 2C Χ Ε over SV. Then the subspace f C | consisting of all pairs (τ, X) for which Ρ(τ)χ = χ is a direct subbundle in % . As follows later (Theorem 1.5), practically every Banach bundle can be obtained by this construction. We note that the triviality of the bundle 3F described above indicates the existence of an operator following of the evolution subspace that is the image of the projection P(r); in other words, an operator U(t, r) on E, depending continuously on the parameters t, f G 3C, that effects an isomorphism of the images of P(t) and P(j). It can be proved that, for a family of operators Τ(τ) ( τ £ 5 ) having zero kernel and complemented images and depending continuously on τ Ε St', there is a continuous family of projections P(r) on the images Im T(T). Therefore, the subspace of the bundle g = 3C Χ Ε consisting of the pairs (τ, Τ(τ)χ) (r G &, χ G E) forms a direct subbundle of it, which we may, for brevity, denote by Im T. The same happens if the kernels of the operators Τ(τ) are identical. Similarly, if all the operators Τ(τ) have range Ε and have complemented kernels, then the kernels form a direct subbundle Ker Τ of the bundle % - 3C X E. Both these facts are true if 3C is paracompact, as may be deduced from Theorem 2.7. Let % be a Banach vector bundle with fibre E, let G be a Banach space for which there is a continuous homomorphism θ of the group GL(E) into the group GL(G). The bundle associated with % is the bundle with the same base and trivializing cover, with fibre G and transition functions θφαβ. If Gi is a subspace of G and is invariant under the operators θφαβ, then to it there also corresponds a bundle with fibre Gl associated with %, which is a subbundle of the previous one. The simplest important example of an associated bundle is %(t) for which G = X(E). Here (eg)A = gAg~l(g e GL(£), A e X(£)). Sections of the bundle X{%) may be identified with endomorphisms of the bundle % . THEOREM 1.3. If JF is a direct subbundle of a bundle % over a parax compact base, then there is a projection from % onto ,Ψ, that is, an endomorphism of % which on each fibre is a bounded projection onto the corresponding fibre of .jF. PROOF. As a preliminary, we note that the set of projections onto a fixed subspace of a Banach space is convex. Therefore, if we can construct the projection in question locally, then we can also construct it globally by means of a partition of unity. Locally this projection can be constructed by using the trivializations described in the definition of the subbundle &. If we choose any projection Ρ from Ε onto F, then Ρ(τ) = φ~ι ο Ρ ο φτ. This proves the theorem. ' Here and below we say, for brevity, "the Banach bundled", or, even more briefly, "the bundled", ρ instead of "the Banach vector bundle §-+3C".

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Under the conditions of the theorem the bundle I decomposes into the direct sum of the subbundle Ψ and of a complementary subbundle "§ . In this case we say that IF is complemented. The next result follows from general facts about bundle spaces (see [59]). THEOREM 1.4. If the base of the Banach vector bundle •% with fibre Ε is a CW-complex and if GL(£) is contractible in the uniform operator topology, then % is trivial. We write Ιλ(Ε) for the Banach space of sequences i*/},^ , *,· S E, with the norm

=f>\\z,L·. THEOREM 1.5. Every Banach bundle % of countable type1 with fibre Ε and paracompact base 3C is isomorphic to a complemented subbundle of the trivial bundle / t = J X i j (£). This theorem was obtained in [109]; its proof is similar to that of a theorem on embeddings of Banach manifolds in [97]. Here the space Ιλ{Ε) can be replaced by a number of other spaces. If, for example, Ε has the property that Ε Θ Ε is isomorphic to a complemented subspace of itself, then we can take Ε in place of Ιχ (£). THEOREM 1.6. Under the conditions of Theorem 1.5 the bundle % Θ 11 is isomorphic to /\. PROOF. According to Theorem 1.5 we may assume that % is a subbundle of 11. We write Ρ(τ) for the projection, which depends continuously on τ, of li(E) onto the fibre I T of I over the point r. Let Lr be the right shift operator on Ι γ. We replace li(E) by the isomorphic space lt 1\{E). Then the isomorphism r % Θ 1\ -*• 1\ of which the theorem speaks can be defined by the formula r (g, gi) = e{® id^ (g) + (Lr® Ρ (ρ (g))) gt + where ex is the first basis vector in lx and ρ is the projection of %. This proves the theorem. 2. Finite-dimensional subbundles. In this subsection we establish certain properties of Banach vector bundles connected with the structure of their finite-dimensional subbundles. THEOREM 1.7. // % is a Banach bundle with paracompact base space 3C then there is a section 8: & -> % that vanishes nowhere. PROOF. We consider a totally ordered family of open sets {Ua} (a G 21), forming a locally-finite trivializing cover of the base X. The section δ is constructed by transfinite induction on a. Let a be a non-limit number and δα_ι a section of % over Wa = U U0 that vanishes nowhere. 0(.r). The image of ( f , 0) is then denoted by [ % ] • If [ % ] is an infinite-dimensional bundle, we may omit the square brackets, because, by Corollary 1 to Theorem 1.9, such bundles can be "pasted together" after factoring by A precisely when they are isomorphic. Thus, the index turns out to be a map, index: \3C, Φ/ΟΕ)] ->• &(&), and, as is easily verified, it is a semigroup homomorphism. From the corollary to Theorem 1.7 it follows that index Τ (Τ: % -> Φ/CE) \ Φ(Ε)) can always be represented in the form (- ITJ), where & is a Banach bundle. Theorem 2.2 carries over, without changing the proof, to families 1 7 ,·: 5V -> Φ/(£)· From this the next result follows. THEOREM 2.4. // Tf 3C ->· Φ ; (£) (/ = 0, 1) are continuous maps of a compact space 3C, if index To = index Tlt and if the group GLCfi1) is contractible, then the families Tt are homotopic. The next result follows from Theorem 2.4. THEOREM 2.5. // GL(£) is contractible and if all infinite-dimensional complemented subspaces of Ε are isomorphic to E, then the set Φι(Ε) \ Φ(Ε) is contractible. PROOF. One shows in the usual way (see [44], [45], [104]) that it suffices to establish that \2C', Φ[(Ε) \ Φ(Ε)] is trivial for an arbitrary CW-complex SC. Let 7,·: 3C -+ Φ,(Ε) \ Φ(Ε) (i = 0, 1) be continuous maps. We claim that index To = index Τχ. Then, by Theorem 2.4, the maps are homotopic. As indicated above, index Ti = (- cP,·), where 5%· is a complemented Banach subbundle of 3C X E. Under the conditions of the theorem s the fibres Ei of the .-9,- are isomorphic to E. Since GL(£) is contractible, the $*j are trivial (see Theorem 1.4) and therefore, index To = index Tt. This proves the theorem. The spaces I (1 < ρ < °°) and c 0 satisfy the conditions of the theorem (see [70], [108], [111]). The index of a family of operators of the class ΦΓ(Ε) will be introduced in §2.3.

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2. Some operator bundles. essarily vector) subbundles of notation we write A instead sets P* = U Φ"(£) C X (£) n>0

In this subsection we consider some (not necthe bundle q: X (E) -*• X (E). To simplify the of q(J) for A C X (E). We also need the and JF~ = U Φη(Ε). It is not hard to establish nΦ(Ε)-^£

(Ε) (Ε)

Here the vertical arrows are locally trivial bundles and the horizontal ones are natural embeddings. We remark that if Φ{η)(Ε) is not empty, then it is homeomorphic to (0) Φ (£). For, as noted in §1.1, the quotient map.g: X (E) -> X(E) is a trivial bundle with fibre C(E)\ that is, ΦΜ(Ε) is homeomorphic to ¥n)(E) X C(E). The spaces Φ{η)(Ε) for different η are homeomorphic, being the preimages of the elements η G Ζ under the continuous homomorphism κ (the index of the operator) from the group of invertible elements Φ(£) C X (E) to Z. It follows from these arguments that φ("\Ε) is homotopy-equivalent to (0) Φ (£·) - GL(£)/GL C (£). (This fact was established in [61]). If now GL(E) is contractible, then Φ ( η ) ( £ ) is a classifying space for GLC(E) (see [59],). One can derive fundamental results of index theory from this (see the Notes on the literature). 3. Global inverses and regularizes. Let A be a Banach algebra and T: & ->· A a continuous map. If T(r) is invertible for each τ €Ξ iV, then Τ~ι(τ) depends continuously on τ [25]. We now suppose that the element T{T) is right invertible for each τ e SV. The question arises whether we can construct an element Τ'(τ) that is a right inverse to Τ(τ) and depends continuously on r. THEOREM 2.7. // 3C is a topological space and Τ{τ) is right (left) invertible for all τ G 3C, then there exists a right (left) inverse depending continuously on τ G JT. PROOF. Since A is a metric space, we can assume that 2C is paracompact. We give the proof for right invertible elements. First we construct Τ'(τ) locally. Let τ 0 G 3C and let So be some right inverse of Τ(τ0). There is a neighbourhood Uo of r 0 such that T(r)S0 is invertible for r G £/0. Then ToOO = So(T(T)Soyl is a local right inverse of Τ(τ) (τ Ε t/ 0 ). Thus, we can construct a locally finite cover SC - U i/ a such that on each neighbourhood t/ a a right inverse Γα'(τ) is defined and depends continuously on r. If {ηα} is a partition of unity corresponding to this cover, then the element Τ'(τ) = Σ ΐ]α(τ)Τ'α(τ) is as required. This proves the a

theorem. DEFINITION 2.3. An operator 5 € # ( £ ) is called a fe/ϊ (ng/zi) regularizer for A & Χ (Ε) ιϊ (BA - \) έ (£) a left (right) regularizer of A goes to a left (right) inverse of q(A). Hence it follows that an operator A has a left (right) regularizer if and only if A G Φ[(Ε) (A G Φ,.ΟΕ')). Then

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135

Β G ΦΓ(Ε) (B G Φ,(£)). If Λ £ Φ(Ε), then each left (right) regularizer Β is also a right (left) regularizer, and Β G Φ(Ε). We are interested in the following question: given a family T(j) of operators, is there a continuous family Τ'(τ) of (left, right, two-sided) regularizers? The next result answers this question. THEOREM 2.8. Let 30 be a paracompact topological space and T: 30 -> ΦΓ(Ε) (Φ{(Ε), Φ(Ε)) a continuous map. Then there is a continuous map T: 30 -»• Φ,(£) (Φ,(£), Φ(£)) swc/z fftaf 7"(τ) is a right (left, two-sided) regularizer for Τ(τ), τ G 30. We give the proof only for the case of a map T: 30 -»• ΦΤ(Ε). For families of Φ,- and Φ-operators the proofs are similar. We consider the map q ο Τ: 30 -»• Χ (Ε). As remarked in § 0, the element q ο Τ(τ) is right invertible in X (E). By Theorem 2.7, there is an element t'(r) G if (£) that is a right inverse to q ο Γ(τ) and depends continuously ο η τ £ ί . Let ρ: έ (£) -> J£ (E) be a Bartle-Graves map (Theorem 1.1). Then the operator Τ'(τ) = ρ ο t'(r) is a right regularizer depending continuously on τ e 3 ' . This proves the theorem. If 3C is compact and T: 2C -> Φ(£), then the index of a global regularizer Τ': 30 -*• Φ(Ε) satisfies the equation index T' - -index T. This follows from index(7T') = index 1 = 0. We can now introduce the index of a family of Φ,,-operators. DEFINITION 2.4. Let 3C be compact and T: 30 ->• ΦΓ(Ε) a continuous map. Then by the index (index T) of the family Τ we mean (-index 7"), where 7" is a regularizer for T. By the theorem proved above, such a regularizer T'\ 30 -*• Φ;(Ε) always exists and its index is defined in the preceding subsection. Index Τ is welldefined, since any two regularizers for Τ are linearly homotopic and therefore have the same index. Analogues of Theorems 2.2, 2.4, and 2.5 hold for families of Φ,.-operators. We pass on to consider some special regularizers for families of Φ-operators. DEFINITION 2.5. A regularizer for a Fredholm operator is said to be left (right) equivalent if it has zero kernel (cokernel). A Fredholm operator has a left (right) equivalent regularizer if and only if its index is non-negative (non-positive) (see [27]). THEOREM 2.9. Let 30 be a topological space and T: 30 -» Φ(Ε) a continuous map. A global left (right) equivalent regularizer exists if and only if Τ is homotopic to a map TV 30 -> Φ(Ε) η GLr{E)

(Φ(Ε)[\σΐι(Ε))

in the class of maps from 30 to Φ(Ε). PROOF. NECESSITY. Let 7": 2T -+ Φ(Ε) be a left equivalent regularizer for T. Then Τ'(τ) is left invertible for all τ G 30. By Theorem 2.6, there is

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a continuous family Τγ: X -* GLr(E) of left inverses to 7". Since the T'(T) are Fredholm operators, the Τχ(τ) are right regularizers for Τ'(τ), and Γ,(τ) G Φ(Ε). Thus, Γ, maps # into Φ(£) η GL,(£). The maps Γ and 7,, being right regularizers for 7", are linearly homotopic in the class of maps from SC to Φ(£). SUFFICIENCY. It is not hard to show that we may assume 3' to be paracompact. Let Tt: SV -»• Φ(Ε) (0 < t < 1) be a homotopy from Γ ο = Γ to 7Ί : S" -> Φ(£) Π GL^CE). The maps q ο Tt: SC -+ j F + are a homotopy from g ο Τ to q ο Γι. By Theorem 2.6, the bundle Φ(£) Π GL,(£) ^ J F + is locally trivial, the map Tx: 3C ^ Φ(Ε) η GL,(£) covers q ο 7\, and therefore, by the homotopy covering theorem (see [58]), there is a homotopy St: 3? -*• Φ(Ε) Π GLr(E) covering q ο Tt and such that 5Ί = Tx. The operators S0(j) differ from T(r) by compact operators. Therefore, a continuous family Τ'(τ) of right inverses to SQ(T), which exists, by Theorem 2.7, gives a global left equivalent regularizer for T. The assertion for right equivalent regularizers is proved similarly. This proves the theorem. COROLLARY. Let 3C be a compact space and T: 9C -» Φ(Ε) a continuous map. There is a global left (right) equivalent regularizer if and only if index Τ = [Ρ] (index Τ = - [Ρ]), where Ρ is some finite-dimensional vector bundle over 3C. We give the proof for the case of a left equivalent regularizer. If there is a global left equivalent regularizer, then, by the theorem, the map T: % -> Φ(Ε) is homotopic to a map Τλ: &•: -> Φ(£) Π G L ^ F ) for which the equality index 7Ί = [P] is obvious, since Coker 7Ί (τ) = 0. Conversely, let index 7^ = [P]. We write 07\ and β,λ for the bundles involved in the definition of index T. Then [ GLi] ~ [ ff>i] = [P]. According to the definition of the group K(3'), this means that there are trivial finitedimensional bundles %χ and % 2 such that $ ι © %\ ~ Ρ © i 2 a n ( 3 Φ(Ε) a continuous map. There is a global equivalent regularizer if and only if Τ is homotopic to a map T1: SC -» GL(E). If SC is compact, this comes to the same thing as the condition that index Τ = 0. Theorems 2.9 and 2.10 admit alternative formulations in terms of the representability of Τ(τ) in the form Q(T) + S(T), where Q(T) is a right, left, or two-sided invertible and S(T) is a compact operator. For example, if T' is a left-equivalent regularizer, then Τ'(τ)Τ(τ) = 1 + 5Ί(τ), where 6Ί(τ) G C(E). By Theorem 2.7 there is a continuous left inverse Q(r) to T'(T).

Then

The next result holds for semi-Fredholm maps. THEOREM 2.11. Let SV be compact and T: SC -> Φ,ΟΕ) \ Φ(£) α continuous map. Then there is a continuous map S: 3C -> K(E) such that Ker(S(r) + T(r)) = 0 for all τ e SC. PROOF. By the remark after Lemma 2.1 we can add to Τ(τ) a finitedimensional operator 5Ί(τ) so that Ker(7Xr) + 5Ί(τ)) = Ν is constant. Then Ιτη(Τ(τ) + 5Ί(τ)) is a complemented subbundle of SC X E. Let § be a complementary Banach subbundle. From the corollary to Theorem 1.7 it follows that % contains a subbundle isomorphic to the finite-dimensional bundle 3C X N. Let φ: SC Χ Ν -*• % be the corresponding embedding. If we write Ρ for a projection onto N, then the operator (£rP + (Τ(τ) + 5\(τ))(1 - Ρ) has zero kernel and differs from Τ(τ) by the finite-dimensional operator S(T) = ψτΡ — Τ(τ)Ρ + S1(r)(l — Ρ). This proves the theorem. 4. Fredholm operators in a pair of Banach spaces. We consider a separable linear topological space A in which Banach spaces Ex and E2 are continuously embedded. On the spaces Eo = El Π £ 2 and Ε = Ex + E2 we introduce the norms | | Ζ | | Ε = max||a;|| E \\Χ\\ = inf (|| ^ || E l + || x2 || E o ), Ο

Ε

i=l,

2

χ=χ,4-χ 2

under which these spaces are Banach spaces and the embeddings E0C E( C Ε C ^< are continuous (ι = 1, 2). We study the algebra Χ (£Ί; £ 2 ) °f bounded linear operators on Ε that take each Eft = 1, 2) into itself. This is a Banach algebra under the norm || A \\x= max \\A \\ ^E-· Clearly, max ( | M IU- -> £· » IM \\E_>E) < || ^4 \\%. In ,Χ we consider the set Φ(Ε1; £ 2 ) °f operators that are Fredholm on Et{i = 0, 1, 2), and also the set . ^ ( £ Ί ; £ 2 ) C Φ(£Ί; Ε2) of operators having the same index on all the Ej(i = 0, 1, 2). We write GL(E1; E2) for the set of operators that are invertible on all the E((i = 0, 1, 2). E

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Μ. G. Zaidenberg, S. G. Krein, P. A. Kuchment, and A. A. Pankov

The set :F(EX, E2) is in a certain sense analogous to the set Φ(Ε) of Fredholm operators on a single space E. For example, if Eo is dense in Ex and E2, then under conditions similar to those of Theorem 2.3 one can prove the exactness of the sequence 0 -+• [£t\ GL (£Ί; E2)] -> [3C, ,f

(£Ί; E2)] — ^

Κ (37) -> 0,

where 3C is compact and index 0 denotes the index of a compact family of Fredholm operators on Eo (see [31], [33]). The structure of the set Φ(£Ί; Ε2) is more complicated. We restrict ourselves here to the case Eo = Ex C E2 = E(EX Φ E2). We assume that dim E2/Ex < °°, where Ex is the closure of Ex in E2. Then the following simple result holds. LEMMA (see [31]). If an operator A is Fredholm on Ex and E2, then κ χ Α < κ 2 Λ. The equality xxA = κ 2Α holds if and only if there is an operator Β 6 %(EX\ E2) such that (AB — I) is finite-dimensional on each of the Et (i = 1, 2). We write I(El; E2) for the set of all pairs (nx, n2) of integers such that there is an operator A with the indices κ tA = nt(i = 1,2). We write Π for the set of pairs of integers {(nx, n2): nx < n2} and Π ο for the set

{«!, n2): nx < 0, n2 > 0}.

It follows from the lemma that I(EX\ E2) C Π if dim E2/Ex < °°. The sets I(EX; E2) for various specific pairs Ex C E2 are studied in [63]. We consider the sequence spaces E2 = I , Ex = I b, where

(

CO

λ

Σ bi\xi\p\1^p < °° and bt is a monotonically increasing unbounded sequence of positive numbers. bn+i

THEOREM 2.12 (see [63]). //lim

= °°, then I(lph,

lp) = Π ο ;

bn+l if lim — — < oo, then 1(1 b, / ) = Π. n-* °°

°n

This theorem implies that an operator that is Fredholm on all the spaces of the scale I ba (0 < α < 1) has index 0 for 0 < a < 1. Further, we can construct an operator for which the indices are different for a = 0 and α = 1. The following result is also established in [63]. THEOREM 2.13. I) If Ex and E2 are Hilbert spaces and Ex Φ Ε2, then either I(EX; E2) = Π ο or I(EX; E2) = Π; 2) I(lp, / ) = Π (1 < p