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Application of methods of the theory of order-bounded operators to the theory of operators in Lp-spaces

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Uspekhi Mat. Nauk 38:6 (1983), 37-83

Russian Math. Surveys 38:6 (1983), 43-98

Application of methods of the theory of order-bounded operators to the theory of operators in Z_p-spaces A.V. Bukhvalov

CONTENTS

Introduction Chapter I. Calculus of order-bounded operators §1.1 Ideal spaces §1.2 Order-bounded operators Chapter II. A criterion for integral representability of linear operators §2.1 Integral operators §2.2 Proof of the criterion for integral representability §2.3 Some applications of the criterion for integral representability §2.4 Calculus of order-bounded operators and the Schrodinger operator Chapter III. Compact operators in spaces of measurable functions §3.1 The problem of majorization for compact operators §3.2 Proof of the majorization theorem for compact operators Appendix. The absence of the property of order-boundedness for singular integral operators References

43 46 47 53 59 59 69 76 78 81 81 83 87 94

Introduction In 1928, at the International Congress of Mathematicians in Bologna, F. Riesz [65] proposed a calculus for continuous linear functionals in the space of continuous functions C[0, 1]. This calculus made it possible to calculate the modulus and the positive and negative component of functionals, which in their properties are in many respects similar to the usual modulus and positive and negative component of a real-valued function. Riesz's construction was based on the consideration of the natural pointwise order relation between elements of C[0, 1] . In the 1930's Kantorovich (see, for instance, the final paper [33]), within the framework of developing a general theory of vector lattices (abstract linear spaces with a "good" order relation), worked out a calculus of order-bounded linear operators on them that contains Riesz's construction as a very particular case. Then he applied it to the solution of abstract functional equations. Next, in the course of the

44

Α. V. Bukhvalov

60's the calculus thus constructed was hardly connected with applications to other areas of analysis (an exception is the interesting approach to the spectral theory of self-adjoint operators in Hubert spaces (see [20], Ch. 9), which however did not lead to new results). Lately the situation has changed substantially: interesting results have been obtained in the theory of operators and the theory of functions for which (so far) no alternative methods of proof have been found. The main aim of the present paper is to give an account of applications of the calculus of order-bounded operators to the theory of operators (including the theory of operators in Hubert spaces and the theory of functions of a real variable), which is addressed primarily to experts in functional analysis and the theory of functions who are not acquainted with the theory of vector lattices and operators on them. All necessary preliminary information is gathered in Chapter I. The chosen range of results is interesting and non-trivial even for I 2 . Every question is presented in a two-step fashion: first the case of the spaces L2 or Lp is considered in detail, and then a brief general formulation is given. The reader who is interested only in L2 or LP may skip over the account of the general case without any loss to the understanding of what follows. No attempt has been made to write a survey of a similar trend, neither in the USSR nor abroad (a very concise account of some results, without proofs and rather obsolete, was given by Lozanovskii and the author in [17]). The range of applications of positive operators to the spectral theory of operators is well known (see, for instance, [81] or the fundamental work by Vladimirov [19] on the single-velocity equation for neutron transfer in nuclear reactors, where the theory of positive operators is applied), and to convex analysis in /^-spaces (see the survey of Kusraev and Kutateladze [47] and the books [ 3 ] , [48]; we remark that the applications in convex analysis are mainly concerned with the Hahn-Banach-Kantorovich extension theorem for operators, which are not touched upon in this paper). Nor do we consider the numerous connections between the theory of Banach lattices with that of Banach spaces, first of all, because in these applications the calculus of order-bounded operators does not play a decisive role (the surveys [15] and [16] and the monographs [51] and [25] are devoted to this topic, although we must mention that [25] contains a wealth of new "ZAresults"). We now briefly describe the content of every chapter. In Chapter I we present the apparatus of order-bounded operators in spaces of measurable functions, in a form that is closest to the needs of the theory of functions and of operators. In 1936 von Neumann [58] raised the problem of a characterization of integral operators in L2. In 1974 this problem was solved by the author [9].

45

Application of the theory of order-bounded operators

The solution is presented in Chapter II. The proof is based on the calculus of order-bounded operators. First the calculus is presented in a form convenient for a wide circle of experts in the theory of functions and functional analysis. Let us quote the result. A linear operator U: L2 -> L2 is called integral if there exists a measurable function K(s, t) (the kernel) such that (1)

(Uf)(s) = [ K(s, t)f(t)dt

for any / £ L2. The integral in (1) is understood in the Lebesgue sense (and so we exclude singular integral operators, operators with kernels that are distributions, operators that are defined not everywhere in L2, etc.) Theorem A. Let U : L2 -> L2 be a linear operator. The following are equivalent:

statements

1) U is an integral operator, that is, U admits a representation (1); 2) if 0 < /„ < / £ I 2 and /„ -> 0 in measure {or equivalently, in the L2-norm), then Ufn -*• 0 almost everywhere. Chapter II also contains applications of the theorem just stated to integral representations of various classes of operators and the resolvent of the Schrodinger operator. In Chapter III we study compact order-bounded operators. In 1976 the renowned mathematical physicist Simon put forward the following conjecture in connection with a study of the Schrodinger operator with a vector magnetic potential. Let U and V be operators in L2. We write It/I < V if I Uf I < V( I / I) for any / E l 2 . Simon conjectured that if V is a compact operator, then so is U. A much more general result was proved in [27] by means of the technique of order-bounded operators (a simplified proof was given in [4]). We state the result of Dodds-Fremlin in the Lp-scale. Theorem B. Let U, V :LP -> Lq be linear operators, 1 < ρ < °°, 1 < q < «. // I Uf I < V( 1/ I) for any f £ W and V is a compact operator, then U is also compact. The most difficult case is ρ = °°. Theorems A and Β are akin in the fact that they are theorems on L2-spaces for which there is no proof by means of the traditional Hubert technique; and apparently such a proof is impossible. Furthermore, although each theorem is an assertion about an individual operator, the proof is carried out by means of embedding this operator in a suitable space of order-bounded operators, which makes it possible to apply the specific operator technique. The paper ends with an Appendix in which we prove that singular integral operators of general form are not order-bounded. The proof of this result shows what problems arise when the calculus of Chapter I is applied in a

46

Α. V. Bukhvalov

specific situation. Besides, the result as such is also of great value in applications to the theory of Sobolev spaces of functions taking values in a Banach space (see [14], where Theorem S2 was first announced; the proof of the theorem is published here for the first time, and so are Propositions SI and S2). The author expresses his sincere gratitude to Professors Ul'yanov, Nikishin, and Olevskii, who conducted seminars at the Moscow State University in which many results of this paper were proved; conversations with them led to the idea of writing this paper. We use the following notation: Ν is the set of natural numbers; C is the set of complex numbers; R" is the η-dimensional Euclidean space; S is the space of all measurable functions of compact support almost everywhere; χΑ is the characteristic function (indicator) of a set A; 1, \T is the function identically equal to 1 on a set T; X £ (t7>) means that the object X has the p r o p e r t y ^ ) ; X ^ {•If') means that X does not have the property (έΡ)\ X* is the space of continuous linear functionals on a Banach space X; E' is the dual space (see §1.1); p' is defined by l/p+ l/p' = 1 if 1 < ρ < °°; • denotes the end of a proof. We also use the following abbreviations: (o) is the abbreviation for the adjective "order"; a.e. means almost everywhere. Definitions of all terms referring to the theory of spaces of measurable functions are given below in Chapter I.

CHAPTER I

CALCULUS OF ORDER-BOUNDED OPERATORS

In this chapter we give an account of the basic facts about order-bounded operators acting in ideal spaces of measurable functions. These are simply p defined objects containing as particular cases the L -spaces as well as the Orlicz, Marcinkiewicz, Lorentz spaces and many others. The basic facts about such spaces are collected in §1.1. The terminology connected with these spaces is simple and allows us to state results in a common form, namely, a convenient language. Here it will be clear whether a result is specific for the W-theory or not (for a non-specialist it may be surprising what a lot of standard W-results admit a common formulation). However, it is not at all for his purpose that we give an account of the foundations of the theory of ideal spaces. Our main aim is to use simple examples to pave


47

the way for more abstract constructions of the theory of order-bounded operators, which is presented in §1.2. As a rule, proofs contained in the easily available monograph by Kantorovich and Akilov [34] are omitted. § 1.1.

Ideal spaces

The task of this section is to establish the terminology connected with an important class of spaces of measurable functions containing all the ZAspaces. We begin with conventions on the character of the measure space. By (Τ, Σ, μ) (possibly with indices) we denote a space Τ with a σ-algebra Σ of measurable sets and a σ-finite measure μ on Σ. The reader not inclined towards such a high level of generality may assume, without making what follows less non-trivial, that (Τ, Σ, μ) is the interval [0, 1] or a domain in R" with the Lebesgue measure. By S(T, Σ, μ), or simply S, we denote the set of all measurable and a.e. finite real-valued functions on (Τ, Σ, ιι) with the usual identification of equivalent functions. Since it is important for the theory of operators that the theory should be applicable also to complex spaces, we remark at once that the condition for functions to be real-valued is nowhere essential: a relevant explanation will be given later. We are interested in the following order relation between functions in S or a subspace of S (say, Lp). For any two functions /, g €Ξ S we set / < g if f(t) < g{t) a.e. It is clear that then S becomes an ordered set and any two functions f.gGS have a supremum f \J g ξ. S and an infimum f f\ g 6 S, defined by (1) (2)

(/ V *)(*) = max (f(t), g(t)), (/ Λ g)(t) = min (f(t), g(t)).

Moreover, for every / G S w e can define its positive part / + , its negative part /_ and its modulus I / I: (3)

/+ = / V 0 ,

/ . = ( - / ) VO,

/ = /+-/_,

| / | = / + + /_,

where, of course, 4

()

1/1(0 = 1/(01-

With every A £ Σ we can associate the projection PA to 5: (5)

(PAf)(t) =

xA(t)f(t).

We have written down the trivial formulae (l)-(5), because they are the basis of almost any calculations in the theory of measurable functions, where they are taken for granted. However, in § 1.2, where we present a calculus of order-bounded operators, the interpretation of the formulae (l)-(5) when / and g are replaced by operators becomes non-trivial.

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Α. V. Bukhvalov

For what follows, we note two further simple properties of the order in S: (6) / < g implies that f+h 0 and λ > 0, then λ / > 0. We now turn to less trivial facts. A set Μ C S is called bounded above if there is a g G S such that / < g for any / G M. Boundedness below is defined similarly. A set Μ C S is called order-bounded (briefly: (o)-bounded) if it is bounded both above and below. This is equivalent to the existence of a g G S such that I / I < g for all / G M. What happens if we wish to calculate the upper bound of an infinite set Μ C £ that is bounded above? If Μ is countable, then g = sup Μ obviously exists and can be calculated by the formula (8) g (t) = sup {/(*): / € M). However, it will be often useful for us to be able to calculate upper and lower bounds of sets of arbitrary cardinality. Note that (8) cannot help us. Firstly, in this case (8) may define a non-measurable function. Secondly, in calculating with (8) we may obtain two measurable but inequivalent functions if we choose different representatives of f(t) in the class of equivalent functions / G M. To see that the difficulties are real and not removable, it is sufficient to consider the interval [0, 1] with the Lebesgue measure. In the first case we can take a set A C [0, 1 ] that is not Lebesgue measurable and consider the set Μ consisting of characteristic functions of all one-point subsets of A. Then the g(t) in (8) is equal to χ^ and is therefore non-measurable. It is obvious, however, that sup Μ = 0. In the second case we can construct an example similarly. It is clear that by means of this construction we can "spoil" the calculation of any uncountable supremum by means of (8). Nevertheless the following theorem is valid, a proof of which can be found in [34] (Theorem 1.6.17). Theorem 1.1. If Μ c: S is bounded above, then Moreover, there is a countable set {fn} cz Μ such (the latter supremum can be calculated pointwise same is true for sets that are bounded below and

g = sup Μ 6 S exists. that sup Μ — sup {/n} by means of (8)). The the infimum.

We give immediately a non-trivial application of Theorem 1.1. For any function / G S we can define, as usual, its support s u p p / = {ί£Τ: }(1)φ0}ζΣ. It is clear that supp / for / G S is defined up to a set of measure zero. For an arbitrary ECS the support of Ε cannot be defined as the union of the supports of the elements of E, for the same reason that makes (8) incorrect. We proceed differently. We consider the set Μ = {%A: A — supp /, / ξ Ε), which is obviously bounded above in S (for example, by 1). Therefore, by Theorem 1.1, g = sup Μ exists. We set by definition supp Ε = supp g.


49

It is easy to show that the set supp Ε ζ, Σ has the natural properties of a support for E: 1) supp / C supp Ε (up to a set of measure zero) for all / €: E; 2) supp Ε is minimal (up to a set of measure zero) satisfying 1). Let us introduce some more definitions. Two elements/, g G S are called disjoint if I / ! Λ \g I = 0. We write fn\ if/„ > fm for n>m. The notation fn \ f means that fnj and / = sup /„ (or fn(t) -*• f(t) a.e.) A similar meaning is attached to fn\ and fn\f. Next, we say that a sequence {/„} converges to / in measure if /„ - > / i n measure on any set of finite measure (we write /„ -»• /(μ)). With the topology of convergence in measure, S(T, Σ, μ) is a complete metric topological vector space ([34], § 1.6). We now come to the consideration of subspaces of S. An ideal space on (Τ, Σ, μ) is a linear subset Ε of 5 such that (9)

if 6 S, g 6 E, I / | < \g I) => (/ 6 E),

that is, with every function Ε contains its modulus function and all functions that are smaller in modulus. A norm II * II on an ideal space Ε is called monotone if (10)

if, geE;

| / | < | g | ) ^ ( | | / | | < \\ g ||).

A Banach ideal space on (Γ, Σ, μ) is an ideal space Ε endowed with a monotone norm with respect to which Ε is a Banach space. All the concepts just introduced are simply stated and natural: practically all spaces in the theory of functions of a real variable are ideal Banach spaces. Such are the classical ZAspaces and the Oiiicz spaces LM, the Marcinkiewicz spacesΜ(ψ), the Lorentz spaces A(i//), also L(p, q), and the Morrey spaces (definitions of the majority of these spaces can be found in [34] or [45]). As was promised, all subsequent main results are connected with Lp, but this concerns only the statements; the proofs use the notions of a vector lattice and a ΑΓ-space, which are more general than that of an ideal space and will be defined at the end of this section. We now return to the concept of an ideal space. A strip in an ideal space Ε is an ideal space F contained in Ε and such that g C f for any Μ C F having the supremum g = sup Μ in E. It is clear that for any fixed Α ζ Σ the set (11)

PAE= {/€£:

supp/c^}

is a strip in Ε (hence the intuitive meaning of the term strip: the strip built over A). Here the operator PA defined by (5) is the projection from Ε to PAE (which is incorporated in the notation). It turns out that, conversely, every strip F has the form (11) for some A. For it is sufficient to set A = supp F. Extensions of the concepts of a strip and a projection to a strip are very important for us in the study of operators.

50

A.V. Bukhvalov

We now list the simplest properties of Banach ideal spaces that are needed in what follows. We write (12) £+ = {feE: />0}. Proposition 1.1 ( [ 3 4 ] , Lemma IV.3.2). If a sequence {fn} converges in norm to f in a Banach ideal space E, then fn -> f in measure. Moreover, there are a subsequence {fnh}, a function r Ε Ε+, and a numerical sequence e h { 0 such that | fnh — f Κ skr.

We do not dwell here on general properties of Banach ideal spaces, but we note that an astonishingly large set of facts depend only on a few conditions with extremely simple formulations, and these facts often have totally nontrivial proofs (see [15], [16], [34]). We say that a norm in a Banach ideal space is (o)-continuous or that Ε satisfies condition 04) (EG (A)) if

(M0)=MII/ n II-*0). We now introduce a convenient notation for majorized convergence almost everywhere in an ideal space E. We say that a sequence {/„} cr Ε (o)-converges to / in £ if/„-»•/ a.e. and if there is a g Ε Ε such that I /„ I < g for all η Ε Ν (we write fn —> /). It is easy to see that (o)-convergence implies convergence in norm if £ Ε (A). Thus, the condition 04) is an abstract analogue of Lebesgue's theorem on majorized convergence. It is clear that Lp Ε (A) for ρ Ε [1, «>) and L°° £ 04). The condition {A) singles out "good" spaces: Proposition 1.2 ([34], Theorem IV.3.3). Suppose that a measure μ is separable (for example, that μ is the Lebesgue measure on a measurable set on the line, or in R", or that μ reduces to countably many point masses; in the latter case all ideal spaces are sequence spaces). A Banach ideal space Ε is separable if and only if Ε satisfies the condition (A). It is clear that the condition (A) is very easily verifiable in specific situations, and in this lies the value of facts like Proposition 1.2. We now pass to the next level of abstraction, by defining general vector lattices. As far as specific spaces are concerned, this class, which is more general than that of ideal spaces, contains perhaps only the space of continuous functions C(K) and therefore would seem to lie aside of the plan of the paper. In fact, however, things are quite different. In the next section we consider spaces of operators acting between ideal spaces (in particular, between Z,p-spaces). Here it is important for us that these spaces are vector lattices. We try to keep the abstract terminology to a minimum by referring all the time to analogies with simple facts about S and ideal spaces. A real vector space £ is called vector lattice if £ is a (partially) ordered set in which any two elements /, g Ε £ have a supremum f \J g and an infimum


51

f /\ g and the linear operations and the ordering are compatible via the axioms (6) and (7). In a vector lattice (3) (but not (1), (2), and (4)) have the obvious meaning. A vector lattice is called a K-space or conditionally complete if any set bounded above has a supremum. Thus, Theorem 1.1 asserts that S is a ΑΓ-space, and then any ideal space is obviously a A^-space. Since the spaces of order-bounded operators to be discussed in § 1.2 are also /T-spaces, we consider from now on only A^-spaces. (C[0, 1] is an example of a vector lattice that is not a A^-space.) We now make a convenient remark that facilitates the verification that a vector lattice £ is a ΑΓ-space. A set Μ C Ε is called directed upwards if for any fx, f2 £ Μ there is an / 3 £ Μ such that f3 > /,, f2. Proposition 1.3. // every subset of a vector lattice Ε that is bounded above and directed upward has a supremum, then Ε is a K-space. Proof. If Μ is an arbitrary set that is bounded above, then we form another set Mi consisting of all suprema of finite collections of elements of M. It is clear that sup Μ and sup Mx exist simultaneously, and if they do, then sup Μ = sup Mi and Mx is directed upwards. • The next result is easy to prove with the help of Theorem 1.1. Proposition 1.4. If Μ is a subset of S that is bounded above and directed upwards, then there is a sequence {fn} a M such that /„ f sup Μ. The notion of being directed downwards is introduced in a similar way and the analogues of Propositions 1.3 and 1.4 for the infimum are valid. An ideal in a .K-space £ is a linear subspace F of Ε such that (I / Κ I g I. / £ E, g 6 F) =>• if 6 F). It is clear that any ideal space is an ideal in S (hence the term). Every ideal is itself a K-space. Our next task is to define analogues of a strip and a projection to a strip in the abstract situation. From now on these concepts are used constantly in proofs. Let £ be a A"-space. Two elements f.gEE are called disjoint (fdg) if | / | /\ | g | = 0. An element / £ Ε is called disjoint from a set Μ C Ε (fdM) if fdg for any g £ M. Disjoint subsets of Ε are defined in the obvious way. We now introduce the operation of forming the disjoint complement, which associates with any MCE the set Md = { / € £ :

(13)

fdM}.

We set M = (A/ ) . If Μ is a subset of an ideal space, then Md is a strip built over T\supp Μ and il/ dd is a strip over supp M. An ideal F in a ΑΓ-space Ε is called a strip if g £ F for every set Μ C F having the supremum g — sup Μ in E. It is not difficult to show that the disjoint complement Md of any set Μ is a strip, hence Μύά is always a strip, which is called the strip generated by M. This operation has no constructive description, a fact that does not prevent it from having important applications to specific problems (see Theorem 2.3 below). dd

d d

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Α. V. Bukhvalov

With any strip F in a A"-space Ε there is connected the canonical projection [F] from Ε to F. For / £ E+ we set (14)

W)

= sup {ge * + : * < / } ·

By the definition of a A"-space, this supremum exists in Ε and by the definition of a strip, [F] (/) £ F. For any / £ £" we set

Obviously, [F] is a linear operator that maps Ε to F and leaves the elements of F in place. Every / £ Ε can be represented uniquely in the form / = g + h, where g £ F and Λ G F d , and g = [F]/, h = [ F d ] / . For it suffices to prove the formula (15)

/ = [F)f + lF*]f

for / £ E+. Since we always have / > [F]f, [F d ]/ and [f ]/ d[Fa]f, we conclude that / > [F]/ + [F d ]/. Suppose that (15) does not hold, that is, / > lF]f + [Fa]f. It is easy to see from the definitions of [F] that (/ — [F]f)d F, so that / — [F)f 6 F* and / > / — I/1]/. Since / — [/"]/ > > [Fa]f, this leads to a contradiction to (14) and proves (15). The existence of the projection (we always mean the canonical projection constructed above) to every strip and the existence of the decomposition (15) is the most important distinctive feature of A"-spaces—there is no need to explain how useful it is to have a large store of projections. It does not matter that these projections, unlike those in (5), as a rule, cannot be written down explicitly, the very fact of their existence gives a lot, as we shall see in Chapters II and III. Later we have to use different inequalities dealing with the ordering in the A^-space of operators. In the case of an ideal space the matter was simple, since in the case of formulae with finitely many linear operations and the operations of taking the supremum and the infimum of two elements the matter reduces to the verification of numerical inequalities, because all these operations proceed pointwise. The verification of a similar kind of inequality in abstract A'-spaces, starting only from the basic definitions, would require cumbersome routine arguments. Fortunately, Yudin's principle of preservation of relations (see [34]) holds in any A"-space; it consists in the following. Suppose that we have two expressions u(xlt ..., xn) and v(xlt ..., xn) formed by means of finitely many linear operations and the operations V and Λ, where all the variables xx xn can take values in an arbitrary vector lattice. Then u ^ ν holds in an arbitrary vector lattice Ε for any values of the arguments if and only if u > υ when we replace xx, ..., xn by arbitrary real numbers (there is, of course, no infinite analogue of this principle). As an example we note that this is the way in which the inequality I (/ V S) — (ft V ?) 1 ^ 1 / — h \ becomes obvious. In what follows we use Yudin's principle without explanation.


53

Finally, let us explain, as was promised at the beginning of the section, what should be done in the case of a space over the field of complex numbers. Let S be the space of measurable functions with complex values. If Ε is a (complex) subspace of S, then the notion of an ideal space is defined as above by means of (9) and the notion of a Banach ideal space by means of (10); the only difference is that the modulus in these formulae has to be defined by the usual formula (4) (the formulae (3) have no meaning in the case of complex values). The collection of all real-valued functions in a complex ideal space Ε form a real ideal space Re(£) (which is a Banach ideal space if Ε is). Next, properties and definitions for Ε are introduced by way of RQ(E). That is how we can define the notions of a strip, the projection to a strip, condition (A) etc. In the next section, where we are concerned with operators acting from one ideal space into another, we have to use the decomposition Ε = Re(E) φ i Re(.E) and the decomposition of an operator on Re(E) into its real and imaginary component (for more details see [81], §§2.11 and 4.1). From now on results are valid both in the real and the complex case, and the proofs are given for real spaces (only the results for the complex case are immediately obtained by passage to Re(£)). §1.2. Order-bounded operators In this section we give an account of the calculus of order-bounded operators in an ideal space, which is basic for our work and was developed mainly by Kantorovich in the middle 30's. A significant contribution was made by his students Vulikh and Pinsker (see [35]) and also by the Japanese mathematician Nakano [56]. As was already noted, we wish to give a meaning to formulae analogous to (l)-(5) in the case of operators. First of all we single out a reasonable space of operators where this can be done. All the operators and functionals to be considered here are linear. Let Ε be an ideal space on (Tu Σ1, μ2) and F an ideal space on (Γ 2 , Σ2, μ 2 ) An operator U : Ε -> F is called order-bounded ((o)-bounded) if it transforms (o)-bounded subsets of Ε into (o)-bounded subsets of F. Before discussing the extent of this class of operators, we give a characterization of such operators. An operator U: Ε ->• F is called positive if from / > 0, /€=£", it follows that Uf > 0. An operator U: Ε -> F is called regular if U = U1— U2 where Ult U2: Ε -> F are positive operators. Proposition 1.5. An operator is (o)-boimded if and only if it is regular. Proof. Every positive operator is obviously (o)-bounded, and a difference of (o)-bounded operators is also (o)-bounded, which proves that every regular operator is (o)-bounded. We postpone the proof of the converse for the time being. •

54

Α. V. Bukhvalov

We denote by L~(E, F) the collection of all (o)-bounded operators. This is obviously a linear set. We introduce an order relation in L~(E, F). We write U > 0 if U is a positive operator. We write U > V if U— V > 0 (in other words, if Uf > Vf for any / Ε Ε+). It is easy to see that we have introduced in L~(E, F) an order relation satisfying conditions (6) and (7). The following theorem due to Kantorovich (which was established by Riesz in [65] for functionals on C[0, 1]) plays the most important role. Theorem 1.2. The set L~(E, F) is a K-space. Moreover, for any U, V G L~{E, F) and f Ε Ε+ the following formulae hold:

(16) (17) (18)

(U V VJ/^supiZ/A + V/,: h, h>0, f = (U/\V)f= inf {Uf, + Vfc fu / 2 >0, f = fx U+f =

(19) (20)

[/„/=-inf |ff|/

(21)

| C / | / = s u p { | ] \Uft\: | / ,

(22)

| C / / | < | i / | ( | / | ) /or a// / € £ .

/„

We begin the proof of Theorem 1.2 with the following lemma. Lemma 1.1. Z,e? £ £>e an ideal space, X a vector space, and U an operator from E+ into X for which (23)

U(f + g) = Uf + Ug for all / , g ζ Ε+

(24)

U(Xf) = Wf\

λ > 0,

ftE+.

Then U admits a unique linear extension to the whole of E. Proof. The uniqueness of the extension is obvious from the formula / = /+—/-·, therefore, it remains to prove the existence. For any / Ε Ε we set Wf = £//+ - i//_. We claim that W is the required extension. To begin with, we verify that for h =f-g,fg&E+, (25)

Wh = Wf — Wg.

It is clear that h+ < / and /z_ < g . Therefore, for / = f—h+ > 0 w e have f = h + +I and g = f-h = h+-h + l = h_ +1. Then'by (23) Wf = Uf = Uh+ + Ul,

Wg = Ug = C/7i_ + Eft,

from which (25) is obtained by subtraction. It is obvious from (25) and (23) that W is an additive operator on E. This, in turn, implies that W(—f) = —Wf for all / Ε Ε, which proves, by (24), that W is homogeneous. •

55


To establish that an ordered vector space X (satisfying (6) and (7)) is a vector lattice it suffices to show that | χ \ = χ \J (—x) exists for any χ G X. For it is sufficient to use the formulae (2b)

/
0 in £ it follows that Ufn —> 0 in F (we recall that in the given ideal space (o)-convergence is majorized convergence almost everywhere). We denote by L^(E, F) the collection of all (o)-continuous operators. Proposition 1.7 ([20 Theorem VIII.3.3]). the K-space L~(E, F).

The space L~(E, F) is a strip in

We conclude this section with some remarks on functionals on ideal spaces and Banach ideal spaces. Let Ε be an ideal space; if we set F = R1 in the definitions above, we obtain the definitions of the space of (o)-bounded (or regular) functionals £ ~ and of the space of (o)-continuous functionals E^. As above, £ ~ is a .K-space and E^ is a strip in E. For any ideal space Ε we define the dual space £" by E' = {g 6 S: \ | fg | ώμ! < oo and supp g a supp E). It is clear that the dual space is an ideal space. It can happen that E' — {0} (for example, Ε — 5(0, 1)) but of course, we are interested in the case when supp Ε — supp E'. If Ε = Lp (1 < ρ < °°), then E' = Lv'. From every g £ E' we can construct a linear functional φί on Ε by the formula (33)

q>,(/) = j /(ί)*(ί), g^E\ It turns out that endowed with this norm £" is a Banach ideal space ([34], Theorem VI. 1.2).

CHAPTER II

A CRITERION FOR INTEGRAL REPRESENTABILITY OF LINEAR OPERATORS In this chapter we prove the criterion for integral representability (Theorem A of the Introduction) on the basis of the calculus of (o)-bounded operators expounded in Chapter I. In §2.1 we discuss general facts on integral operators (see also [34], §XI.l and [44]); also the history of the question on integral representability and the statement of the criterion itself. In §2.2 we give a proof of Theorem A. In §2.3 we discuss some consequences of the criterion, and in §2.4 we consider the question of the integral representability of the resolvent of the Schrodinger operator. §2.1. Integral operators Throughout this chapter Ε denotes an ideal space on (7\, Σχ, μ χ ), F an ideal space on (Γ 2 , 2 2 , μ 2 ), and (Τ, Σ, μ) the product of the measure spaces (Tlt Σΐ7 μχ) and (Γ 2 , Σ 2 , μ 2 ). As we have already said, the reader may think that Ε - F = L2{0, 1). An operator U: Ε -» F is called integral if there is a μ-measurable function K{s, t) {t G T1, s G T2) such that for any / G Ε the value of g = Uf is the function (1)

·

g(*) = the kernel of U.

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Α. V. Bukhvalov

We emphasize that the operator (1) must be defined on the whole of Ε and that the integral in (1) is understood as the usual Lebesgue integral. This excludes operators that are densely defined in V by (1) (so that, for example, Theorem III.5.1 of [8] on the representation of the resolvents of elliptic operators by means of the Green's function cannot, at least formally, be regarded as a result on the integral representation of the resolvent; however, see [70], where, for elliptic equations with smooth coefficients this representation is extended to the whole of Lp). It also excludes operators in which convergence of an integral is understood in the sense of one summation method or another; i.e. singular integral operators (where the integrals are interpreted in the sense of the principal value either pointwise almost everywhere or in the Z p -metric, 1 < ρ < °°) which in fact fail to be integral in the sense of (1), but are even order-bounded with values in the broadest ideal space S (see the Appendix). To the same case there belongs formally the classical "integral representation of the resolvent" in the theory of ordinary differential operators like Theorem II.3.1 of [49], where a limit passage in the ZAmetric occurs (the situation is considered explicitly in [6]). For many potentials an integral representation of the resolvent is still possible. Nevertheless it seems that for certain classes of potentials the question remains open. We also observe that a limit passage in L2 can actually lead to a loss of the property of integral representability. For the operator of the Fourier transformation tip: L x (—oo, oo) -»-L°°(—oo, oo) is an integral operator, whereas the Fourier-Plancherel operator obtained by passage to the limit in L2 is not (o)-bounded from L2(-°°, °°) to S(-°°, °°) (see the discussion after Theorem 1.2), hence is not an integral operator (see below). Finally, operators whose kernels are distributions do not fall under the definition (1); in fact, with a suitable definition of the class of distributions any operator can in this case be regarded as integral. It was the discussion between von Neumann and Dirac (in 1932) about the possibility of applying the theory of integral equations to the investigation of the operators of quantum mechanics that led von Neumann to the problem of a description of the class of integral operators in the sense of (1) (see [59], where there is a criticism of the approach of Dirac, who used his 2 delta function as the "kernel" of the identity operator in L and constructed "kernels" of differential operators by means of the derivatives of the delta function). Nowadays, of course, the problem has lost much of its urgency, since we have a mathematically justified apparatus of distributions and the technique of equipped Hilbert spaces appeared. But historically it was precisely the discussion that led to the fundamental work by von Neumann [58] in which the problem of finding all operators in L2(0, 1) that are unitarily equivalent to a self-adjoint operator (see Theorem 2.2 in this paper) was solved and the problem of finding necessary and sufficient conditions for an operator in L2(0, 1) to admit an integral representation (1) was first stated.


61

The last problem was solved by the present author in 1974 (see Theorem A in the Introduction and Theorem 2.1 below) on the basis of the calculus of order-bounded operators expounded in Chapter I. Apparently, at that time it was the first case when this calculus was applied to solve a new problem of analysis that in its statement is not connected with order (other examples of a similar kind will be given in Chapter III). Theorem 2.1 was published with a proof in [9] and announced in [10]. The announcement attracted the attention of Zaanen, the notable Dutch expert in the theory of integral operators and vector lattices, who gave a talk on the subject at a conference on the theory of ordered spaces and operators in them at Oberwolfach (1977; see [31]) and also gave to one of his students Schep the problem of finding a proof of Theorem 2.1 (the paper [9] was translated into English only in 1978); he succeded to do this based on hints in [10] (Schep's dissertation [77] and the paper [78] are devoted to a full proof of Theorem 2.1). In [79] and [80] Schep obtained a number of new applications of Theorem 2.1 to the integral representation of some classes of operators (we mention that Theorems 4.2 and 4.3 of [78] and Theorems 2.2, 2.4 (the implications (i) •*> (ii) «*• (iii)), and 3.2 of [79] are contained in the author's paper [11]). Schep's proof of Theorem 2.1 differs only in details from the one in [9]. Abstract generalizations of Theorem 2.1 (giving no new information on operators in ideal spaces) were obtained in [18], [24], and [30]. An application of Theorem 2.1 to the investigation of operators in L2 is given in [41]. In [73] there is another criterion for integral representability, which reduces easily to Theorem 2.1 [80) (but so far there is no way of reducing Theorem 2.1 to the result of [73]). Theorem 2.1 is quoted without proof in the monographs by Kantorovich and Akilov [34] (Theorem XI. 1.1) and Korotkov [39] (Theorem III. 1.1). In 1978 there appeared the monograph [72] of Halmos and Sunder,which revived the interest in the theory of bounded integral operators in L2 and V (see, for example. [73] - [ 7 5 ] , [42], where there are answers to many questions raised in [72] ; partially their results are reflected below). Halmos and Sunder were not acquainted with the author's papers [9] and [10] ; therefore, in the introduction of [72] they named the problem of finding necessary and sufficient conditions for an operator in L2 to be integral as being the key open problem in the theory of integral operators in L2. As we said already, the problem had been solved, which was mentioned in a detailed review of [72] by Zaanen [32] , where there is also a detailed analysis of the proof of Theorem 2.1 and where it was specially emphasized that it was the technique of the theory of (o)-bounded operators, not the Hubert space technique, that led to a proof of Theorem 2.1 and a solution of the problem of a criterion for integral representability. Having of necessity dwelt on priority questions, we now return to considering integral operators. First we list their elementary properties.

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The definition (1) preassumes that the following two conditions hold. 1) For any / € Ε and almost every s £ T2 the integrand in (1) is summable, that is,

(2)

j | K(s, t)f(t) | άμώ) < oo

for almost all s G T2. 2) For any / G Ε the function (1) belongs to F; that is, if F = L2, say, then it is required that

(3)

0 a.e. 3) // U acts from a Banach ideal space Ε into a Banach ideal space F, then U is continuous. Proof. By virtue of (2), 1) and 2) are obvious consequences of Lebesgue's theorem. 3) According to the closed graph theorem, it is sufficient to check that g = Uf from /„ -+ / in the norm in Ε and Ufn -»• g in the norm in F it follows that g - Uf. By Proposition 1.2, there is a subsequence /„ —> f in E. Then i//n -*-Uf a.e. by 1). Using Proposition 2.1 again we find that Ufnh -*~S in measure. Therefore, g — Uf. * For Banach ideal spaces satisfying certain conditions (in particular, Lp) Proposition 2.2.3) was proved by Banach ([7], Theorem V.9); for the general case, see [22]. Remark. In view of the preceding account, any integral operator from a Banach ideal space Ε into S is continuous (5 is considered with the topology of convergence in measure), hence, takes the unit ball of Ε into a set that is bounded in measure. The question of a characterization of operators from W into Lq in terms of the kernel here aside of the topics of this paper (in this connection, see Kantorovich's sufficient conditions in [44], [34], §7 and §XI.3; for necessary and sufficient conditions, see [43], they are based on ideas connected with the so-called Schur device). Let us return now to Proposition 2.1. Proof of Proposition 2.1 for the case Ε = I 2 ( 0 , 1), F = 5(0, 1). It is sufficient to establish that if U G L~(L2, S) then (8) holds for all / G L2, f > 0. We fix such an / and consider the set Μ = {g: \ g | ^ /}, which appears in the formula (20) of Ch. I for the calculation of the modulus. For almost all s (9)

j | K(s, t) | f(t)dt « sup {j K(s, t)g(t)dt: g ζ Μ},

where the supremum on the right-hand side of (9) is pointwise. To verify that (9) holds it is enough to remark that for any s the function (10) satisfies \gs I < / and

(H)

g.(t) = sign(K(s, t))f(t)

j I K(s, t) | f(t)dt = j K(s, t)gt(t)dt.

The formula (28) of Ch. I asserts that (12)

| U | / = sup {Ugi g


65

but the supremum on the right-hand side of (12) is the supremum in 5. A repeated error in proofs of (8) is that the authors do not notice the difference between the character of the suprema in (9) and in (12). It was explained in § 1.1 that, generally speaking, information about the pointwise supremum says nothing about the supremum in S, even in the case of the Lebesgue measure. Here we have to use the specific character of our set U(M). Since L2(0, 1) is separable, there is a countable everywhere dense subset {gn } in M. We claim that (13) (14)

j | K(s, t) | f(t)dt = sup {j K(s, t)gn(t)dt: η 6 Ν} a.e.; sup {Ug: g 6M} = sup {Ugn: η ζ Ν},

from which (8) follows immediately. If g £ 1 is arbitrary, there is a sequence gjih —>-g in norm, and then, by Proposition 1.1 a subsequence gnh

—•> g in L2. By Proposition 2.2, Ugnh m

m

->-£/g a.e. Therefore, η

from which (14) follows at once. But does (13) follow from (15)? No. since on the right-hand side of (9) (which is equal to the left-hand side of (13)) there stands a pointwise supremum over an uncountable set of g €Ξ Μ. whereas (15) is an inequality that holds a.e. and the excluded set of measure zero depends on g. It is necessary to modify somewhat the preceding argument. We fix an 5 for which (2) holds (this is so for almost all s) and consider the function gs defined by (10). Suppose that gnh ->~g, in norm. By Proposition 1.1 we can choose a subsequence gm —» ge. Then by means of Lebesgue's theorem we find that (Ugm)(s) -> (Ugs)(s) (we recall that s is fixed; here we have used (2)). Hence and from (11) we obtain (13). • Corollary. An integral operator U (see (1)) is positive if and only if K(s, t)>0 μ-a.e. Also, U = 0 if and only if K(s, t) = 0 μ-a.e. An elementary proof of the corollary, which is not based on the more difficult Proposition 2.1, is given in [ 3 4 ] . Let us now dwell on the question of the adjoint of an integral operator. To begin with consider the case Ε — F — L2. Along with (1). we consider the transposed operator (16)

(U*g)(t) = ( K(s,

(this acts on functions defined on T2): we cannot guarantee that U* is defined for all g G L2, but even if it is defined for some g G L2, we cannot be sure that U#g G L2. By identifying L2 with the space of linear functional 2 2 on L , we regard the adjoint operator U* as an operator in L .

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Α. V. Bukhvalov

Proposition 2.3. // U is a regular integral operator in L2, then so is U* and If = U*. The assertion of Proposition 2.3 may be valid even without the assumption that U is regular (for example, when the kernel is symmetric or skewsymmetric [44]), but it does not hold in general ([72], Example 7.2; [ 3 9 ] , Example II. 1.17). An example of a normal integral operator in L2(0, 1) whose adjoint is not integral is constructed in [75], and then U* does not act i n i 2 ; ([72], Theorem 7.5), which solves a long-standing problem (see, for example, Problem 11.2 in [72]). Now we make some remarks on the case of general ideal spaces and Banach ideal spaces. The reader interested in the L2-case only may omit them. Let U be an integral operator from a Banach ideal space Ε into a Banach ideal space F. The transpose U# is defined by the same formula (16). It can be shown (see, [28], Proposition 2.1) that for the adjoint operator U*\ F* -+ E* we always have U*(F^) C £ ~ (this is obvious when U is regular). If we identify £ ~ with E' and F^ with F' by Proposition 1.8, we may regard U* as an operator from F' to E'. There is an analogue to Proposition 2.3 according to which U*: F' -»• E' is an integral operator and U* = U# if U : Ε -> F is regular. We now turn to the main problem of the chapter: when does a linear operator in L2 admit an integral representation? Apparently it was first proved by von Neumann in [59] that the identity operator / in L2(0, 1) does not admit an integral representation. Even by this example we can illustrate the difference between the conditions 1) and 2) of Proposition 2.2, which arose from two totally equivalent statements of Lebesgue's theorem on the passage to the limit under the sign of integral. In the case of the identity operator / the condition 1) holds trivially, it is simply a tautology, while 2), of course, does not hold, since it is an easy matter to find a sequence of characteristic functions /„ on [0, 1 ] such that /„ -> 0 in measure, but not almost everywhere. This argument is the simplest way to prove that / is not an integral operator. It turns out that the property 2) of Proposition 2.2 characterizes integral operators. Theorem 2.1 (Bukhvalov [9], [10]). Let U: L*(Tlt Σ 1 5 μ^ -*-&{ΤΛ Σ,, μ,) be a linear operator. The following assertions are equivalent: 1) U is an integral operator, that is, U admits a representation (1); 2 2) if 0 < /„ < / Ε L (n G N) and fn -+ 0 in measure, then Ufn -> 0 a. e.; 3 ) i / 0 < / n < / e i 2 ( n e N ) and/„->() in norm, then Ufn -> 0 a.e.; 4) U satisfies the following two conditions: a) if μ^Αη) ->-0 (An £ Σ ) 2 and μι(νΑη) < °° then f/(X4n) -+0a.e.; b) if 0 < /„ < / G L (n G N) and fn -* 0 a.e., then Ufn -* 0 a.e.


67

A proof of Theorem 2.1 will be given in §2.2, but first we discuss the statement. The implication 1) =* 2) is trivial and was proved in Proposition 2.2. The principal equivalence in Theorem 2.1 is 1) ·*> 2), which is nontrivial just in the interesting direction: the implication 2) =* 1) is a sufficient condition for the operator to be integral. The equivalence 2) •**· 3) is evident, and so is 2) =* 4). The assertion 4) emerged as a refinement of the main statement 2). The principal condition in 4) is a), which means that a condition similar to 2) holds, but only for characteristic functions, which may prove to be useful in applications. Thus, only 4) =» 1) in Theorem 1 needs to be proved, and this will be done in §2.2. If μχ is a discrete measure, then all operators admit an integral representation [34], therefore, the discrete case is of no interest. The cases Tt = T2 = [0, 1] and 7~i = Γ2 = R", both with the Lebesgue measure, are basic. The reader may choose here whatever level of generality he prefers: the arguments in the proof do not simplify because of that. Theorem 2.1 is valid for any pair of ideal spaces (in particular, for operators from LP to Lq): in this degree of generality it is given in [9], [10], [34] ; no essential change arises. We give the statement to facilitate further references. Theorem 2.1' ( [ 9 ] , [10]). Analogues of Theorem 2.1.1), 2), and 4) hold for a linear operator U acting from any ideal space Ε to any ideal space F {and if Ε is a Banach ideal space satisfying the condition (A), then also the analogue of 3)). Let us discuss briefly the ways to prove Theorem 2.1. First of all, looking at 2), which has the form of a specific continuity property of U, one may think that one theorem on integral representation for functionals or another might be helpful. One could wish to fix an 5 £ T2 and introduce a functional on L2 by setting φ,(/) - (Uf)(s)· But the very notion of the value of a measurable function (or, more accurately, of a class of equivalent functions) at a point s is not defined. There is a very non-trivial theorem of von Neumann-Maharam on the existence of a lifting, which allows us to give a meaning to the notion for functions belonging to L°°, but in W with ρ < °° no lifting can exist in principle (see [26]). Suppose, however, that even this difficulty proves to be surmountable. A new one would appear when verifying that φ5 is continuous on L2 for almost all s, consisting in the fact that the set of measure zero in 2) that is excluded by the condition Ufn ->· 0 a.e. depends o n {/n}· Suppose again that we have surmounted this difficulty. Then we 2 can apply Riesz's theorem on the general form of a functional in L and write

Μ) = f g.{t)f(tWiit)*

68

A.V. Bukhvalov

or, setting K(s, t) = gs(t),

(Uf)(s) = φ,(/) = j K(s, As a function of two variables, K(s, t) is, in general, non-measurable, but, as we shall see in §2.3, it can be corrected and made measurable. Now we observe that by Riesz's theorem I | gs(t) | 2 *(ΤΧ, Σ υ μ,) = = L2(0, 1) , F — L 2 (0, 1) (see [53], where, in fact, a more general result is stated), but in the case of operators from L2 to L2 it characterizes not all integral operators, but only regular ones. The first very simple though non-trivial idea in the proof of Theorem 2.1 when we are concerned with operators in L2 is that we must necessarily go beyond the framework of L2 and consider operators from L2 to S. In this case, as we already know, all integral operators are regular so that in the further proof of Theorem 2.1 we can use Theorem 2.3 with F = S. In this (and in fact in a more general) setting Theorem 2.3 was proved in [9] (see also [29]). The proof of Theorem 2.3 is based on ideas of Lozanovskii, although it differs in some details. In reading any result in integral representation, including Theorem 2.1 it is natural to ask where and how will the author construct the representation itself and whether the representation is perhaps based on some difficult result which removes the whole complication. In this case the answer to this question must be in the negative. A specific result on integral representations is used in proof of Theorem 2.3 and is contained in Lemma 2.1 below; it has the form of a theorem of Dunford-Pettis type and has been known for a long time, at least for the case of [0, 1 ] with the Lebesgue measure. A proof based on the Bochner integral was given in [9] (in [77] the proof of Tmeorem 2.3 is reduced altogether to the Radon-Nikodym theorem, which is somewhat longer than ours). Here we give quite an elementary proof of Lemma 2.1 using only the general form of a linear functional on L1 whose idea is similar to that of proofs of more general facts in [11]. Lemma 2.1. Every linear continuous operator U: V-(TU Z l t μι)->-+L°°(TZ, Σ 2 , μ 2 ) is integral. Proof. Let Μ be the set of functions of the form

(Μι (4|), μ2 (Bi) < oo), where Al{)Ai=0

{ΐφ /). We set

m

φ (L) = 2 λ, J V (%Ai) άμζ. It is easy to see that the linear functional φ on Μ is well-defined.

Moreover,

so that φ is continuous on Μ endowed with the Ll(T, Σ, μ)-ηοΐτη. Since Μ X is dense in norm in L {T, Σ, μ) [34], φ can be extended by continuity to a bounded functional on V-(T, Σ, μ) for which we retain the earlier notation.

71


Then there is a function Κ 6 Ζ,°°(Γ,,Σ, μ) such that 0. Remark. Lemma 2.3 has a simple meaning from the point of view of the analogy between the theories of functions and of (o)-bounded operators that was developed in Ch. I. The idea is that Vx plays the role of the function identically equal to 1. Then the operators Ό f\ Vn are simply truncations of U, therefore, Lemma 2.3 indicates that the truncations converge to the operator. Proof. 1) To begin with we prove the required relation for U = f ® g, where f'GL% and g €Ξ F+. It is clear that (U Λ Vn)(f) = ^f(t)(f'(t)g(s)

Λ nlT(s, ί))ί μ ι (ί)

(this follows, for example, from Proposition 2.1, but can easily be verified directly). Since f{t)g(s) /\ n\T \ f(t)g(s), we obtain the required result from the theorem of Beppo Levi and the formula (29) of Ch. I. 2) Next we verify that if W e K(L2, F), W > 0, and W /\ Vt = 0, then W = 0. We take an arbitrary operator V > 0 of the form /' ® g. Then according to what has just been proved, sup (V f\ Vn) = V. Hence,

w Λν =w

Λ

sup (ν Λ vn) = sup (w Λ νη

Λ

to = o,

since 0 < W /\ Vn < η (W f\ Fj) = 0. By the definition of K(L2, F) as the strip generated by the operators of the form /' ® g, we conclude that W - 0 (we have proved that W is disjoint to K(L2, F)).

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A.V. Bukhvalov

3) We now turn to the proof of Lemma 2.3 for an arbitrary U. By Theorem 1.2 there is an operator F = sup (U f\ Vn) ^ U in the iT-space L~(L2, F). It is clear that sup (F n Λ V) = sup [(F n Λ £7) Λ VI = sup (F n Λ £0 Λ V = V. We set W = (J7 — V) Λ Vi > 0. Then

(Vn Λ V) + W = (F n + if) Λ (V + W)^

F n + 1 Λ U,

therefore, F = sup (U Λ Fn) = sup (17 Λ Vn+i) > sup (F f\ Vn) + W = V + W. Consequently, W < 0. Thus, W = 0, that is, ( 0. We have to prove that U admits an integral representation. By Lemma 2.2, there is a kernel Kn(s, t) for Un = U /\ F B that is,

(Unf)(s) = \Kn(S, tMWtoit), Since 0 ^ Un\, by Proposition 2.1 0 ^ Kn\. Un f U. We set

f € LK

According to Lemma 2.3,

K(s, i) = η

(a priori, this function may assume the value +°° on a set of positive measure). As follows from Ch. I (29), Unf f £7/, / e L%. By the theorem of Beppo Levi we find that U is an integral operator with the kernel K(s, t) (which then automatically has compact support a.e.). 2) Let U be a regular integral operator (1). We claim that U Ε K(L2, F). Since U is regular U G L^(L2, F). Without loss of generality we may assume that U > 0. We set Kn(s, t) = K(s, t) /\ nl(st t) and let Un be the integral operator with kernel Kn(s, t). Then for all / Ε L%

Since Vn e £(L 2 , F) and 0 < Un < 7W, we see that [/„ G ^(L 2 , F) because every strip is an ideal. By the theorem of Beppo Levi Un\ U (here we again use Ch. I (29)) and then U G K(L2, F) by the definition of a strip. • Corollary. The set of regular integral operators from L2 to F is a strip. Although some information on the structure of the strip of integral operators has been lost in the corollary, it is very useful (the corollary was rediscovered in [50]). The non-trivial part of the proof consists in showing


73

that regular integral operators form an ideal in the sense of the theory of ΛΓ-spaces (not operators). Hence, we single out this result as a separate criterion, which often works in specific situations (see §2.4). Theorem 2.4. An operator U: L2 -*• S is integral if and only if there exists an integral operator V > 0 such that \U\ < V. We now return to the proof of Theorem 2.1. We start with three auxiliary facts. The first of them is a necessary and sufficient criterion for subsets of S to be (o)-bounded; it has been known for a long time in the theory of ΛΓ-spaces, even in a more general form [35], Theorem V.3.31). Proposition 2.4. A set Μ 0 we shall arrive at a contradiction. Thus, /„ f + on A. Applying Egorov's uniform convergence theorem to the sequence {1/(1 + /„)} we find that there is a set Β C Α, μλ{Β) > 0, such that for any C> 0 there is an « c G Ν such that fn(t) > C for all t G Β and all η > nc. We set C = m (m G N) and construct a sequence nx < n2 < ... < nm < ... such that /„(?) > m for all i G " when η > nm. Now we set \ m = \jm ->· 0. We see that λ,η/η (ί) ^ ^ Xmm ^ l , i 6 i , m £ N , which contradicts to the fact that by (+) we must have ^mfnm -+0a.e. • Proposition 2.5 ( [ 2 0 ] ) . u e L~(L2, S).

Remark.

If U : L2 -+ S satisfies Proposition

2.2,1), then

Instead of L2 we can take any ideal space E.

Proof. Since any / G L2 can be represented in the form f = f+ -/_, it is sufficient to prove that U carries any set L = {g 6 Li1: 0 ^ g ^ /} (/ g LI) into a bounded set in S. We use Proposition 2.2 to verify that Μ = U(L) satisfies (+). Let \ n -> 0, gn = Ufn, 0 < / „ < / . Then \nfn -*· 0 a.e. and Ι λ η 1 / η < sup Ιλ,,Ι/G L2. Consequently, by hypothesis, Xngn — \nUfn =

74

Α. V. Bukhvalov

Lemma 2.4 ([9]). Let {Pm} be a sequence of subsets of S(Tt, Σ { , μ2) such that Pm •=> Pm+J (m G N) and inf {g: g 6 Pm) = Ofor all m. Then there is a sequence of functions {gn} such that (18) (19)

Hm_gn(s) = 0 a.e.; for allm there is an n{m): {gn: η > n(m)} cz

Pm.

Proof. Since inf {g: g £ Pm) = 0, for any m there is a sequence {hkm} of the form hhm = g™ Λ · · · Λ «Γ', where g|m> g P m i s such that hhm \ 0 as k -» oo. By the diagonal sequence theorem ([21], Theorem VII.4.5) there is a sequence kx < k2 < ... < km < ... such that fchmm ->0 a.e. as m ->• °°. We define the sequence {gn} as follows. First we write down the elements £ί'\ · · ·, ^'«Mrnrng hhii then the elements ^ ! ) , . . ., gfs' defining Λ h j 2 , and so on. inf {gn' n^ 1} ^ feftmm for any / Ε Ν if m > m,. Now let m -* °°. We see that inf {gn: n^ 1} = 0, hence lim gn(s) = 0 a.e. (19) follows from the construction and the fact that Pm m decreases. • 1

Proof of the implication 4) =* 1) in Theorem 2.1. By Proposition 2.4 it follows from b) that U G L~(L2, S) (we cannot assert that U Ε Z,~(I 2 , I 2 ) , therefore, a proof using the technique of (o)-bounded operators within the framework of L2 is impossible). By Theorem 2.3, the set of integral operators from L2 to 5 is the strip K(L2, S) generated by the finitedimensional integral operators. From the basic properties of the operator calculus of Ch. I it follows that there is a projection operator Pr from the ϋΓ-space L~(L2, S) onto the strip K(L2, S). We set W = t / - P r U. The operator W G Z,~(L2, S) satisfies 4). a) in Theorem 2.1. Since the action of Pr does not have a constructive description, this would be difficult to obtain directly from the fact that U has this property. Here again we have to use Theorem 2.3. For, by Theorem 2.3, Pr U has an integral representation, and then a) holds. Now both U and Pr U satisfy a), and then so does W as their difference. Suppose that U is not an integral operator. Then W Φ 0, by Theorem 2.3. We derive from this a contradiction. According to the definition of the projection onto a strip, W is disjoint to K(L2, S), in particular, to all finite-dimensional integral operators. We fix an arbitrary set C ζ Σ χ, Mi(C) < °° and claim that (20)

W(Xc) = 0.

By what has been said, in particular, this is true f o r / ' = % c G L2. Using Ch. I (31) for the infimum of operators, we now see that (21)

(|W


75

We set It is clear that Pm=> Pm+1 (m Ε Ν). From (21) we obtain that

hence, inf Pm = 0 for all m € N. By Lemma 2.4 there are sequences {An} and {Bn } such that (22) C = An[)Bn, Anf}Bn = 0, (23) 1 π η ( | Η Ί ( χ Α η ) + μ 1 (5 η )1 τ ,) = 0 a.e., (24) Now (23) follows from (18) and (24) from (19). Since | W(%A ) I I W | (5u n ), by (23) (25)

1 ί Ξ ( | ^ ( χ Λ η ) | + μ 1 ( 5 η ) 1 τ , ) = 0 a.e.

Since Xsn ^Ξ Zc € Lz, using (24) and the property a) of W we find that W(%Bn) -+-0 a.e. Consequently, We now choose an s G T2 for which both the limit relations (25) and (26) hold. From (25) we infer that Hm | W(%A ){S) | = 0. If we compare this with (26), we see that W(%c)(s) = 0 for such s, and then W(%c) = 0 a.e., that is, (20). We have now reached the end of the proof, therefore, the moment has come to explain where the difference between convergences a.e. and in measure is used (that is, between the conditions 1) and 2) of Proposition 2.2). By going over to a subsequence, we could achieve by (24) that %B ->-0 a.e. But then (23) would be violated (because in (23) there stands the lower limit and not the usual limit a.e., where the transition to a subsequence would not spoil anything). It remains to complete the proof of the fact that W = 0, by extending by continuity (20), which is already proved, to all functions in I 2 , which is possible because W Ε L^(L2, S). • Remark 1. In the second part of the proof we have used some ideas of Nakano's paper [57], in which positive bilinear forms on /^-spaces are studied. Remark 2. We mention one more peculiarity of the proof of Theorem 2.1: the calculus of (o)-bounded operators has enabled us to reduce the problem to the case of the spaces L1 and L°° (Lemma 2.1). This is not accidentalmany results are proved this way (for example, Theorem 3.1 below).

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Α. V. Bukhvalov

In conclusion we remark that Theorems 2.3 and 2.4 are true for any ideal spaces Ε and F (see [9]): in the definition of K(E, F) one must take the operators /' ® g with f'EE'.gEF. §2.3. Some applications of the criterion for integral representability There are some specific classes of operators whose elements all admit integral representations. In certain cases of particular pairs of Banach ideal spaces all continuous operators admit integral representations. Theorems of this type were established in the 30's: such are the classical theorems of Dunford-Pettis, Kantorovich-Vulikh, and Gel'fand (for a survey of earlier results, see [35], Ch. VIII; also [34], §XI.l and [11]). All the results stated here are not in the most general form (see [11]). Theorem 2.5. Let U be a linear operator from ^(7",, S l f μχ) to L*(TS, Σ,, μ^. The following assertions are equivalent: 1) U is a Carleman operator, that is, integral with kernel satisfying (4); 2) iffn ->· 0 in the norm of L2, then Ufn -» 0 a.e.; 3) there is a function g £ S+ such that (27)

| Uf | < || / || g for all f € L\

in other words, the image of the unit ball of L2 is (o)-bounded in S. Proof. The implications 1) => 3) =* 2) are obvious, 2) =* 3) follows from Proposition 2.4. Finally, 2) =» 1) is evident from Theorem 2.1.3). • (27) clearly is the most convenient for verification (see [38]). We state a more general result in the Lp-scale, which can be obtained with the help of Theorem 2.1'. Theorem 2.5'. Let U: LP(TU Σ 1 ( μ^ -+S{T2, Σ,, μ,) (1 < ρ < °°) be a linear operator. The following assertions are equivalent: 1) U is an integral operator and for almost all s £ T2 2) iffn^-0 in the norm of Lp, then Ufn -> 0 a.e.; 3) there is a function g £ S+ such that \Uf\^\\f\\g

forallf£Lp.

For ρ = °° Theorem 2.5' does not hold: the identity operator in I°°(0, 1) satisfies 2) and 3), but is not integral. Theorem 2.5 generalizes to the case when for the domain we take a Banach ideal space having the property (A). From Theorem 2.5' we infer, in particular, that any bounded operator from LP (1 *ζ ρ < °°) into L°° is integral (the case ρ — 1 is Lemma 2.1). By passing to the adjoint operator, we infer from Theorem 2.5' that any bounded operator from L1 into Lp (1 < ρ < °°) is integral.

77


Next we note some simple consequences of Theorem 2.1 connected with integral representability of compositions of operators (for the case of more general spaces, see [53], [9]). Proposition 2.6. I) If U is a regular integral operator in L2and V £ L~(L2, S), then W = VU is integral. 2) // U e L^(L2, L2) and V : L2 -+ S is an integral operator, then so is W = VU. Proof. Let us verify, say 1), by means of Theorem 2.1. Let 0 0 a.e., and since U is regular, the sequence {Ufn} is (o)-bounded in Z,2, that is, Ufn —> 0 in L2. Since V £ L~(L2, S), it follows that VUfn -+ 0 a.e. • We observe that the assumptions of Proposition 2.6 cannot be weakened substantially. In [42] and [74] there are examples constructed independently of two integral (compact) operators in L2(0, 1) whose composition is not integral (there and in [40] there are many other interesting results connected with integral representability of products of operators). By means of Theorem 2.1 it is also easy to prove that if an "integral" operator is generated by a non-measurable kernel, then the kernel can be replaced by a measurable one. Proposition 2.7 ([9]). Let Ε be an ideal space and Φ(5, Ο a (possibly nonmeasurable) function such that for any f G Ε the function g(s) = Ι Φ(«, t)f(t)d\ix{t) is μ2-α.β. of compact support and μ Then there is a μ-measurable function K(s, t) such that for any / G Ε

(Uf)(s) = J O(s, ί)/(ί)ίμι(ί) = j W for μϊ-almost

all s {the set to be excluded

depends,

t)f{t)dH{t) in general,

on f ' ) .

Proof. Just as in the proof of Proposition 2.2 we verify by Lebesgue's theorem that U satisfies Theorem 2.1.2). • Corollary ([23]). If under the assumptions of Proposition 2.7 the measure space (7\, Σ 1 ? μ!) is separable and supp Ε = 7Ί, then K(s, t) = Φ(5, t) for all s and μγ-almost all t. Remark.

The corollary does not extend to general measure spaces.

New proofs of Proposition 2.7 and its corollary were found in [73]. We note that both results can be interpreted in terms of the theory of random processes (see Theorem 1 in [67]). We also note that Proposition 2.7 gives a way of obtaining results on integral representability for operators in the sense of (1) from results on the representation of operators by means of vector-valued functions [12].

78

A.V. Bukhvalov

At the beginning of §2.2 we mentioned that it is easy to construct a compact operator in L2 that is not integral. Fremlin's example of a positive compact operator in L 2 (0, 1) that is not integral [71] is much more complicated. Using this fact we can easily construct a sequence of finitedimensional positive operators Un in L 2 (0, 1) such that Un -> U in norm. This shows that the set of positive integral operators in L2 is not closed under the operator norm. To see this, it is sufficient to take averaging operators Pn in L2 with respect to the partition of [0, 1] into η equal intervals and to set Un = PnU. Since U is compact, Un -*· U in norm and Un > 0, since U > 0 and Pn > 0. §2.4. Calculus of order-bounded operators and the Schrodinger operator Up to the beginning of the 70's the calculus of (o)-bounded operators (in particular, positive operators) was regarded as hardly connected with the study of the operators of mathematical physics, since the basic object of these investigations is a self-adjoint unbounded operator in L2 and is never (o)-bounded. In the early 70's it became clear that the notion of a positive operator (in the sense of Ch. I, not of the theory of self-adjoint operators in a Hilbert space) is important at least to establish that the Schrodinger operator is essentially self-adjoint (see [64], §X.4, where positivity in our sense is called pointwise positivity; see also [37], §2). In [37], §3 and [62] classical theorems on integral representability like those stated after Theorem 2.5 are applied to the problem of integral representation of the resolvent of the Schrodinger operator. The more general theorems of §2.3 and of the papers quoted there apparently give more general statements in this direction. In [37] and [62] only representation by means of kernels having some additional degree of summability in comparison with (1) are treated. It seems that in the literature there is no theorem on integral representation of the resolvent by general kernels. We also mention that so far in applications to the Schrodinger operator the calculus of Ch. I is not used in essence, except for the simplest inequality I Uf I < f/(l / I) for positive operators (see Ch. I, (22)). The method of studying the Schrodinger operator with a magnetic vector potential proposed in [2] shows that many properties of the resolvent of an operator with a vector potential are not worse than those of the resolvent of the corresponding operator without the vector potential. Here we establish this by means of Theorem 2.4 for the property of integral representability of the resolvent (the results to follow are due to the author and are published here for the first time). For the basic facts about the Schrodinger operator we refer the reader to [63], [64], and [37]. Let Δ be the Laplacian in R", V an ordinary potential (the operator of multiplication by a real-valued function V) and a = (al, ..., an) a magnetic vector potential such that a,- 6 L\0Z{W).


We write (28) (29) (30) (31)

79

Ho = - Δ ; Η - - Δ + F; # 0 ( a ) = (iV + a) 2 ; #(a) = (iV -f a) 2 + F .

The operator (31) is called a Schrodinger operator with a magnetic vector potential. For a rigorous definition of (30) as an essentially self-adjoing operator on CQ, see [ 2 ] , 849-850. We consider the following conditions on V: a) V is an //0-bounded operator with the relative bound a < 1; b) F = V+— V_, where F+ 6 £Joc a n d F_ is Z/0-bounded in the sense of forms with the relative boundary a < 1. By Theorems 2.4 and 2.5 of [2] in both cases the operator (31) is well defined and essentially self-adjoint on CJ° (for b) in the sense of forms), and (32)

I e-'H(a) ι 0. The modulus and the inequality in (32) are to be understood in the sense of the calculus of Ch. I (note that in this section we deal, of course, with a complex space L2). The proof of (32) is based on the following inequality due to Simon ( [ 2 ] , Theorem 2.3): (33) μ-ΐϊ,ικι^-ΐΗ,-,,ι^ 0. Now exp(/A) is a positive integral operator in Z,2(R"): (34)

(eM/) (σ) = -~^2

\ e~ | s ~ o t ''"f (s)

ds

R"

(see, for instance, [64], Theorem IX.29). The operator εχρ(/ΥΔ) defined for all t € R1 by means of the same formula (34) (with t replaced by it), with the only difference that in this case the integral in (34) is understood in the proper sense only for / €= L1 Π L2, while for an arbitrary / £ L2 we need the same passage to the limit as for the Fourier-Plancherel transform, not only fails to be integral (no unitary operator can be integral, by Theorem 2.2), but is also not (o)-bounded (for an (o)-bounded operator in L2 that is integral on an ideal space dense in L2 extends to an integral operator on the whole of I 2 ) . In what follows we do not tie ourselves to specific restrictions on the potential V, but we state some general requirements axiomatically. There are many distinct sufficient conditions for these requirements to hold simultaneously (see [ 2 ] , [37]). Thus, we assume that V is such that the operators (29) and (31) are well defined as essentially self-adjoint on CQ (form-sums are admitted) and that (32) holds. Note that (32) even presupposes that exp(-///) > 0 (see also [37] for conditions ensuring the last inequality). The promised result on integrability of the resolvent of //(a) then follows.

80

Α. V. Bukhvalov

Proposition 2.8. If for some μ € C the resolvent (μ/ — Ε)'1 of Η is an integral operator (for sufficient conditions see [ 3 7 ] , Theorem 1.4), then the resolvent (λ/ — if (a))" 1 of //(a) is an integral operator for any λ satisfying Re λ < inf a(H). Remark. As will be clear from the proof, an analogue to Proposition 2.8 remains valid if the word "integral" is replaced everywhere by "regular integral". Lemma 2.5. Let U be a closed operator in L2. //(μ/ — U)-1 is (o)-bounded in L2 and (λ/ — U)'1 is (regular) integral, then so is (μ/ — U)"1. Proof. It is easy to show that every (o)-bounded operator in L2 is (o)-continuous (if /„ -H* 0, then it is easy to construct a function r £ L\ such that | /„ | ^ e,,r, where ε η ->-0 (see [20], Theorem VII.6.2), therefore I tf/n Κ

βη I I? |(r) ¥1 0 in L2). We write down Hubert's identity (μ/ — [Τ)"1 — (λ/ — U)'1 = (μ - λ)(λ/ — ϋ)-ι(μ1 — U)~K

It now remains to apply Proposition 2.6. • Lemma 2.6. If for some μ Ε C f/ze resolvent (μ/ — /f)- 1 o/ // is an integral operator then (KI — H ) ' 1 is an integral operator for all λ satisfying Re λ < inf a(H). Proof. In what follows we use several times the formula (35) below. Let U be a self-adjoint operator and Re λ < inf a(U). Then (see [63], [64])

(35)

(λ/-£/)-'= - [ ο

eae-tudt.

The integral can be understood in the following weak sense: if we denote by ( · , ·) the scalar product in L2(R"), then for any /, g G L2(R")

( ( j e*e-«/ dt) f, g) = j i In our case it follows from (35) that for all relevant λ the operator (λ/ — Η)-1 is (οHounded in ΖΛ (36)

| (λΙ-ΗΓ1

j 0 and denote by [/(λ)] the projection to the strip generated by /(λ). For η = 2l (I G N) and i = 0, 1, . . . , « - 1 we set

*K>(*£)]-[> (4)]. >»-sVi··

85


It is clear that /„ f . Next, it can be shown (see [20]) that

therefore ! / - / „ I < (l/n)e, hence /„ f /. • Remark. The last inequalities in the proof of Lemma 3.2 cannot be derived from Yudin's accordance principle, since they involve projections to a strip. Axiomatic proofs of these inequalities are not simple, therefore, we refer the reader to [20] for details. We now turn to the proof of Theorem 3.1 for the case ρ = °°, which is singled out in the following proposition. Proposition 3.3. The space of all (o)-bounded compact operators from g / / " ( F L Σ 1 ( μχ) toL (T2, Σ2, μ 2 ), 1 < q < °°, which we denote by $t is a strip in the K-space L~(L°°, Lq). Remark. Since a strip is always an ideal, we obtain the required case of Theorem 3.1, but the statement of Proposition 3.3 contains additional information. For example, if U Ε Z.~(Z,°°, Lq) is compact, then so is \U . As we mentioned in §3.1, this is not true in L~(L2, L2). Proof. We write Β = {f £ L*>: f > 0, || / |U < 1}. To prove that U is compact it is obviously sufficient to establish that the set U(B) is totally bounded, that is, it has a finite e-net in Lq for any ε > 0. We break the further proof into several steps. 1) Suppose to begin with that an operator W ξ 38 is given as W = U+ V, where U and V are disjoint, that is, | U | Λ 1^1 = 0. We claim that U 6 £ ρ ( Γ 2 ; Σ 2 , μ2) is compact. As was mentioned at the beginning of the section, to prove that U is compact it is sufficient to verify that U is compact in measure. We go over to the adjoint operators, which act from L"'(Tt, Σ 2 , μ2) to LV'{TX, Σ Χ , μ^. It is easy to see that It/*I < V* and V* is compact. Since the measures have been assumed to be finite, 2>°(Γ2, Σ 2 , μ2) cz L9'(T2, Σ 2 , μ 2 ). We consider the restrictions of U* and V* to L°°(Tt, Σ,, μ 8 ), which we denote by U and V. As before,

87


obviously It/I < V and V is compact from L°° to U". Since 1 < p' < °° by Proposition 3.3 t/ is compact from L°° to U". Going over once again to the adjoint operator, we find that U - U* is compact from £ ρ (7\, Σ χ , μχ) to ^ ( Γ ϋ , Σ 2 , μ2). This implies that the operator U on IP is compact in measure. • We now mention possible generalization of Theorem 3.1 and related results. Theorem 3.1'. Let Ε and F be Banach ideal spaces such that E' and F satisfy (A). Let U, V : Ε -+ F be linear operators. If I U\ < V Ε Z~(£, F) α«ύ? F is compact, then so is U. Remark. In [27] and [4] it was required that (A) holds in the K-space Ε*, instead of the weaker condition Ε' Ε (A); to preserve the result we had to require that V Ε L~(E, F) instead of V > 0. We note that in [27] and [4] £ could be a more general space than a Banach ideal space; for example, one could take Ε = C[0, 1] for V > 0. As was already mentioned, the conditions Ε' Ε (A), F Ε (A) cannot be weakened even in the Zp-scale. Nevertheless the following curious theorem was obtained in [4]. Theorem 3.2. Let Ε be a Banach ideal space, let U and V operators in Ε such that If/I < V, and let V be compact. restrictions on Ε U3 is compact (there is an example of an that U3 — 0, but U2 is not compact). If Ε Ε (A), then U2

be linear Then without any operator such is compact.

In comparison with Theorem 3.1, Theorem 3.2 gives new information in the I p -scale for ρ = 1: if U, V: L1 -> L\ It/I < V, and V is compact, then t/2 : V- -> L1 is compact; and for ρ = °°: if t/, V : L°° -+ L°°, It/I < V, and F is compact, then U3: V° -+ L°° is compact. Curious results of the same kind as Theorems 3.1 and 3.2 were obtained in [5] for the property of weak compactness, but this, is less interesting from the point of view of applications. APPENDIX

THE ABSENCE OF THE PROPERTY OF ORDER-BOUNDEDNESS INTEGRAL OPERATORS

FOR SINGULAR

In [13] and [14] the present author developed an apparatus suited to the P solution of the following problem. Let X be a Banach space. By L (X) (1 < ρ < °°) we denote the Banach space of all measurable X-valued functions f such that

f II = ( j II 1(0 is finite.

\\pdt)1/P

88

Α. V. Bukhvalov

Functions of the form

(i)

( Σ / k ® **)(*) = Σ h(t)xk ft=l ft=l

P

are dense in L (X) in norm. Let U be an operator in Lp. We define a natural extension U of it to X-valued functions of the form (1) by the formula

(2)

&(j]fh®xk)=

Σ

(Ufh)®xh.

If this operator is bounded in the Lp(X)-norm, it admits an extension by continuity to the whole of LP(X), which we also denote by U. The continuity of U often gives interesting inequalities for the case of functions of many variables, which allows us to impose sometimes conditions of a different kind on different groups of variables. The continuity problem for U goes back to the classical work by Marcinkiewicz and Zygmund [55] in which some positive results were obtained. However, by no means every operator U has U bounded in LP(X). An elegant result in this direction is due to Kwapien [36] for the case when i7 = ^ is the Fourier transform in L2(—oo, oo): 0

(3)

j

{Uef)(s)=

K(s-t)f(t)dt,

is an integral operator from Ε to S. We assume that for any / G Ε the following limit in the sense of convergence in measure exists in S:

(4)

Uf=limUJ,

that is, the integral in the sense of the principal value. We assume that the following conditions hold, which are natural for the theory of singular integral operators:

K(lt) = λ~ηΚ (t)

0, t £ R");

\K{t-s)-K(t)\dt^B0);

Hl>2|s|

(7)

j

\K(t)\dt=+oo;

there is a p, 1 < ρ < °°, such that for any α, β, 0 < a < β < °°,

(8)

j

|Z(i)| p di6m/m then | s , - 1 1 > | s0—11 — | s 0 - s t \>6m/2m by virtue of (11). Thus, for (12)

02|So-.sI|

Now we give a lower estimate the minuend in (13). By (10), mes (B (s0; 6 m ) Π CA) = m e s (B (*0; 6 m )) - mes (Af]B (s0, 6 J (s 0 ; 6 m ) ) - ( l — v m ) mes where c m = mes (5(s 0 ; 6 m )). Using (8), we then have (15)

(

\K{so-t)\lA{t)dt^ \ (s 0 - t) | | X B ( e o ; 6 m ) (i) - χ Α (f) (ymcm)1"''

The expression >K(so-t)\vdt)

V(P-l)

92

Α. V. Bukhvalov

does not depend on m, therefore, we can take ym such that 0< T m |s o -i|3>6 m /m

Then by (15), (16)

j

j

\K{sa-t)\lA{t)dt^

|5o-il>em/m

\K(t)\dt-i.

l>U|^l/m

Substituting (14) and (16) in (13) we obtain (17)

l^e(/».J(Si)|>

J

\K(t)\dt-B~l.

By (7), the right-hand side of (17) tends to + °° as m -> °°. Now l/ m , i o l < for all m and s0. If (9) does not hold, then there are a set Al C A of positive measure and a number Ν such that (18)

\U(fn.J{s)\^N

a.e. on / ^ for all m and s 0 (the s e t to be excluded, generally speaking, depends on m and s0). We take an arbitrary density point s0 of ^4j and, consequently, of A. By (7) there is an m such that \K(t)\dt-B-i>N + i. Then from (17) we obtain that | sx — s0 | < 6m/2m for | U^m,,^) |> > iV + 1. Since £/E(/m,s0) does not depend on ε for ε < 6m/2m, it follows that •

(19)

I # (/«...)(«)

for almost all 5 such that | s — s 0 | < 6m/2m. Since 50 is a density point of Ax, we may assume that mes(^ a f] -S(s0; 6m/2m)) > 0. Comparing (18) and (19) we conclude that for almost all s £ Ax f] B(s0, bml2m) which is a contradiction and proves the theorem. • A direct application of U to / m , s does not give anything, since the pointwise supremum over an uncountable set turns out to be unbounded and may be greater than the supremum with respect to the order relation. In this connection we recall that a similar procedure could not be carried out even in the case of ordinary integral operators (see Proposition 2.1). Let us clarify in more detail some specific applications of Theorems A.I and A.2. Let U be a singular integral operator with a Calderon-Zygmund kernel (see [69], Ch. II): = v.p.


93

Such operators are bounded in Lp, 1 < ρ < °°. Let X be a Banach ideal space of functions depending on a variable v. We consider the space LP[X] with a mixed norm consisting of all measurable functions g(t, v) for which the norm II g \\Lvm = II II g(t, ν) ||^, Β \\LV% i s finite (see [34]). If X is separable, then LP(X) = LP[X]; but in general, LP(X) C LP[X]. In this case U acts by the formula

^

/

,

dt,

v)

R"

that is, i/ acts out like U and "does not do anything" on v. Such singular integral operators are important, for example, in the investigation of function spaces with a dominant mixed derivative, which were first introduced by Nikol'skii. Theorems A.I and A.2 show that an equivalent uniformly convex norm can be introduced in X if X is a Banach ideal space and U is bounded either in LP[X] or in LP(X) (that is, X is not necessarily reflexive, but in addition satisfies a strong geometric condition similar to the Clarkson inequalities in Lp). In particular, if X — LM is an Orlicz space, then U is bounded in LP[LM] if and only if LM is reflexive (the sufficiency can be proved by means of the author's theorems on interpolation by the method of Ovchinnikov in spaces with a mixed norm). In LP[X] it is also natural to consider singular integral operators that act on both t and v. They are useful in the investigation of Sobolev and Liouville spaces in the case of mixed norms. Such operators do not fall within the framework of Theorem A.I. However, a very simple modification of the proof Theorem A. 1.1) A. 1, 2) bearing in mind that the Hubert singular integral operator DO

= v.p. j

- ^

dt

is an isomorphism of any Banach space in which it acts, gives the following result on the two-dimensional Hubert transform (it can easily be generalized to the «-dimensional case). Let σο

oo

(Hzg)(s, u>)=v.p. J j -S--±-g(t, $—

Ϊ

v)dtdv.

— oo — oo

P

If H2 is bounded in L [X], then we can introduce an equivalent uniformly convex norm in X. Hence, using Lizorkin's theorem on the boundedness of H2 in Lp[Lpi], 1 < ρ, ρΎ < oo, we can prove that H2 is bounded in L2[LM] if and only if LM is reflexive. Added in proof. For new approaches connected with the integral representation of operators, see the paper by Weis [82]. In connection with the material of the Appendix on vector-valued extensions of operators we refer to [83], [85]. We mention that a number of earlier results of the

94

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author in [13], [14], were rediscovered in [83], in particular, Theorem A.I and a number of related results on singular integral operators. It is proved in [83], using Pisier's ideas [89], that there is an equivalent uniformly convex norm in X if the Hubert singular integral operator is bounded in P L (X), and [85] contains an example of a uniformly convex Banach space X P such that the Hilbert singular integral operator is not bounded in L (X) (1 < ρ < °°). We note that so far it is not clear whether the method of [83] can be applied to general singular integral operators covered by Theorem A.2. Finally, the Banach space in [85] and [89] cannot be isomorphic to any Banach ideal space. For the case of Banach ideal spaces the only known result seems to be that of the author, which can be proved by means of his interpolation results in spaces of vector-valued functions. Proposition. Let U be a singular integral operator with kernel of Calderon-Zygmund type, 1 < ρ < °°. P 1) The vector-valued extension U is bounded in L (LM), where LM is an Orlicz space, if and only if LM is reflexive. 2) The vector-valued extension U is bounded in Lp(L(r, q)), where L{r, q) is a Lorentz space, if and only if 1 < r and q < °°. References [1] Yu.A. Abramovich, Injective hulls of normed structures, Dokl. Akad. Nauk SSSR 197(1971), 743-745. MR 44 # 7257. = Soviet Math. Dokl. 12(1971), 511-514. [2] J.E. Avron, I.W. Herbst, and B. Simon, Schrodinger operators with magnetic fields. I. General interactions, Duke Math. J. 45 (1978), 847-883. MR 80k:35054. [3] G.P. Akilov and S.S. Kutateladze, Uporyadochennye vektornye prostranstva (Ordered vector spaces), Nauka, Novosibirsk 1978. MR 80i:46007. [4] CD. Aliprantis and O. Burkinshaw, Positive compact operators on Banach lattices, Math. Z. 174 (1980), 289-298. MR 81m:47053. [5] and , On weakly compact operators on Banach lattices, Proc. Amer. Math. Soc. 83 (1981), 573-578. MR 82j:47057. [6] N.I. Akhiezer and I.M. Glazman, Teoriya lineinykh operatorov ν gilbertvom prostranstve (Theory of linear operators in a Hilbert space), Nauka, Moscow 1966. MR 34 # 6527. [7] S. Banach, Theorie des operations lineaires, reprint of first ed., Chelsea Publ. Co., New York 1955. MR 17-175. Translation: Kurs funktsional'nogo analizu, Radyank'ka Shkola, Kiev 1948. [8] Yu.M. Berezanskii, Razlozheniya po sobstvennym funktsiyam samosopryazhennykh operatov (Decomposition in eigenfunctions of self-adjoint operators), Naukova Dumka, Kiev 1965. MR 36 # 5769. [9] A.V. Bukhvalov, On integral representation of linear operators, Zap. Nauchn. Sem. LOMI 47 (1974), 5-14. MR 53 # 3767. [10] , An integral representability criterion for linear operators, Funktsional. Anal, i Prilozhen. 9:1 (1975), 51. MR 51 # 8882. = Functional Anal. Appl. 9 (1975), 45.


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Leningrad Polytechnic Institute Received by the Editors 21 October 1982

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