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Complex interpolation of compact operators mapping into lattice couples/Vorepaaridesse kujutavate kompaktsete operaatorite kompleksne interpolatsioon.

After 44 years it is still not known whether an operator mapping one Banach couple boundedly into another and acting

compactly on one (or even both) of the "endpoint" spaces also acts compactly between the complex interpolation spaces generated

by these couples. We answer this question affirmatively in certain cases where the "range" Banach couple is a couple of lattices on

the same measure space.

Key words: functional analysis, complex interpolation, compact operator, Banach lattice.

Kusimus, kas Banachi (ruumide) paaride vahel tegutsev tokestatud lineaarne operaator, mis tegutseb uhel

(voi isegi molemal) lahteruumil kompaktselt, tegutseb kompaktselt ka nende ruumide poolt genereeritud

komplekssete interpolatsiooniruumide vahel, on pusinud lahtisena juba 44 aastat. Artiklis on vastatud sellele

kusimusele jaatavalt juhul, kui operaatori sihtpaar on teatavaid loomulikke eeldusi rahuldav (uhel ja samal

mooduga ruumil tegutsevate) Banachi vorede paar.

1. INTRODUCTION

All Banach spaces in this paper will be over the complex field. The closed unit ball of a Banach space A will be denoted by [B.sub.A]. For any two Banach spaces A and B, the notation T : A [??] B will mean, just like the usual notation T : A [right arrow] B, that T is a linear operator T defined on A (and also possibly defined on a larger space) and it maps A into B boundedly. The notation T : A [??] B will mean that T : A [??] B with the additional condition that T maps A into B compactly.

We will write A [??] B when A is continuously embedded with norm 1 into B, and A [??] B when A and B coincide with equality of norms.

For each Banach couple (or interpolation pair) [??] = ([A.sub.0],[A.sub.1]) and each [theta] [member of] [0,1], we will let [[A.sub.0],[A.sub.1]][theta] denote the complex interpolation space of Alberto Calderon [3]. We also let [A.sup.o.sub.j] denote the closure of [A.sub.0] [intersection] [A.sub.1] in [A.sub.j] for j = 0,1. The couple ([A.sub.0], [A.sub.1]) is called regular if [A.sup.o.sub.j] = [A.sub.j] for j = 0,1. The spaces [A.sub.0] [intersection] [A.sub.1] and [A.sub.0] + [A.sub.1] are Banach spaces when they are equipped with their usual norms (as e.g., on p. 114 of [3]).

For any two fixed Banach couples [??] = ([A.sub.0], [A.sub.1]) and [??] = ([B.sub.0], [B.sub.1]), the notation T : [??] [??] [??] will mean that the linear operator T : [A.sub.0] + [A.sub.1] [right arrow] [B.sub.0] + [B.sub.1] satisfies T : [A.sub.0] [right arrow] [B.sub.0] and T : [A.sub.1] [??] [B.sub.1]. The notation [??] * [??] will mean that every linear operator T : [A.sub.0] + [A.sub.1] [right arrow] [B.sub.0] + [B.sub.1] which satisfies T : A [??] B also satisfies T : [[[A.sub.0] ,[A.sub.1]].sub.[theta]] [??] [[[B.sub.0],[B.sub.1]].sub.[theta]] for every [theta] [member of] (0 , 1). The notation (*.*) * [??] for some fixed Banach couple [??] will mean that [??] * [??] for every Banach couple [??]. Analogously, the notation [??] * (*.*) for some fixed Banach couple [??] will mean that [??] * [??] for every Banach couple [??].

Some 44 years ago, Calderon [3] proved that (*.*) * [??] for all Banach couples [??] which satisfy a certain approximation condition. Since then it has been established that [??] * [??] for a large variety of other different choices of [??] and [??]. (See, e.g., the 12 papers and website referred to on p. 72 of [7], and [7] itself.) However, we still do not know whether [??] * [??] holds for all choices of [??] and [??], i.e., whether "(*.*) * (*.*)".

In this paper we shall add to the library of known examples of couples [??] and [??] satisfying [??] * [??] in the context of spaces of measurable functions. We shall use the terminology lattice couple to mean a Banach couple [??] = ([A.sub.0], [A.sub.1]) where both [A.sub.0] and [A.sub.1] are complexified Banach lattices of measurable functions defined on the same [sigma]-finite measure space.

Cobos et al. [4, Theorem 3.2 p. 289] proved that [??] * [??] whenever both [??] and [??] are lattice couples, provided that [B.sub.0] and [B.sub.1] both have the Fatou property, or that at least one of [B.sub.0] and [B.sub.1] has absolutely continuous norm. Subsequently, Cwikel and Kalton [8, Corollary 7 part (c) on p. 270] generalized this result by showing that [??] * (*.*) for any lattice couple [??].

In this paper we shall obtain a different generalization of the above-mentioned result of [4], namely we will show that (*.*) * [??] for every lattice couple [??] satisfying one or the other of the same conditions imposed in [4]. In fact, some other weaker conditions on [??] are also sufficient. Roughly speaking, as indeed the reader might naturally guess, our approach is to take the "adjoint" of the above-mentioned result [??] * (*.*) of [8], using arguments in the style of Schauder's classical theorem about adjoints of operators. But this is apparently not quite as simple to do as one might at first expect.

In forthcoming papers we plan to extend our main result (*, *) * [??] to more general lattice couples and non-lattice couples B, including some which are rather close in some sense to the couple ([l.sup.[infinity]]([FL.sup.[infinity]]),[l.sup.[infinity]] ([FL.sup.[infinity].sub.1])). We recall (see [9], or [5]) that (*.*) * ([l.sup.[infinity]] ([FL.sup.[infinity].sub.1]),[l.sup.[infinity]]([FL.sup.[infinity]])) if and only if (*.*) * (*.*).

Pustylnik [21] recently obtained a very general compactness theorem which has some overlap with our result here.

2. A RATHER GENERAL ARZELA-ASCOLI-SCHAUDER THEOREM

In this section we describe the result which will play the role of Schauder's theorem for the proof of our main result.

Let us recall that a semimetric space (X, d), also often referred to as apseudometric space, is defined exactly like a metric space, except that the condition d(x,y) = 0 for a pair of points x,y [member of] X does not imply that x = y. (However, d(x, x) = 0 for all x [member of] X.) Each semimetric space (X, d) gives rise to a metric space ([??],[??]) in an obvious way, where X is the set of equivalence classes of X defined by the relation x ~ y [??] d(x,y) = 0.

Here are three definitions and three propositions concerning an arbitrary semimetric space (X, d). The definitions are exactly analogous to standard definitions for metric spaces, and the propositions are proved exactly analogously to the standard proofs of the corresponding standard propositions in the case of metric spaces, or by invoking those standard propositions for the particular metric space ([??], [??]).

Definition 2.1. Let B(x, r) denote the ball of radius r centred at x, i.e., for each x [member of] X and r > 0, we set B(x, r) = {y [member of] X : d(x,y) [less than or equal to] r}.

Definition 2.2. The semimetric space (X, d) is said to be totally bounded if, for each r > 0, there exists a finite set [F.sub.r] [subset] X such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Definition 2.3. The semimetric space (X, d) is said to be separable if there exists a countable set Y [subset] X such that [inf.sub.y[member of]Y] d(x,y) = 0 for each x [member of] X.

Proposition 2.4. If (X, d) is totally bounded, then it is separable.

Proposition 2.5. (X, d) is not totally bounded if and only if for some r > 0 there exists an infinite set E [subset] X such that d( x, y) > r for all x, y [member of] E with x [not equal to] y.

Proposition 2.6. (X, d) is totally bounded if and only if every sequence [{[x.sub.n]}.sub.n[member of]N] in X lias a Cauchy subsequence, i.e., a subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] which satisfies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

The following theorem obviously contains the classical theorem of Schauder, and it is a simple exercise to show that it also contains the classical theorem of Arzela-Ascoli. After obtaining it we learned that, even though it generalizes these two very important theorems, it is itself merely a special case, a "lite" version, of considerably more abstract results presented by Bartle in [1] (cf. also e.g., [19]) and which, as explained in [1], have their roots in earlier work, mainly of R. S. Phillips [20], Smulian [23], and Kakutani [14]. However, it seems easier to give a direct proof of this theorem than to deduce it from [1]. Furthermore, we learned that essentially the same theorem had also been obtained independently, apparently slightly before us, by Eliahu Levy. His proof in [16] is perhaps better than the one to be given here, and Dr. Levy and I have since refined it to obtain a quantitative result [10].

Theorem 2.7. Let A and B be two sets and let h : A x B [right arrow] C be a function with the properties that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

Define [d.sub.A] ([a.sub.1], [a.sub.2]) := [sup.sub.b[member of]B] [absolute value of h([a.sub.1], b) - h([a.sub.2], b)] for each pair of elements [a.sub.1] and [a.sub.2] in A. Define [d.sub.B] ([b.sub.1], [b.sub.2]) = [sup.sub.a[member of]A][absolute value of h(a, [b.sub.1]) - h(a, [b.sub.2])] for each pair of elements [b.sub.1] and [b.sub.2] in B. Then (A, [d.sub.A]) and (B, [d.sub.B]) are semimetric spaces and

(A, [d.sub.A])is totally bounded if and only if (B, [d.sub.B]) is totally bounded. (3)

Proof. It is obvious that (A, [d.sub.A]) and (B, [d.sub.B]) are semimetric spaces. For the proof of (3), because of the symmetrical roles of A and B, we only have to prove one of the two implications. Suppose then that (A, [d.sub.A]) is totally bounded. By Proposition 2.4, there exists a countable subset Y of A which is dense in A. Let us show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

The inequality "[greater than or equal to]" in (4) is obvious. For the reverse inequality, given any [b.sub.1] and [b.sub.2] in B and any arbitrarily small positive [epsilon], we choose a [member of] A such that

[d.sub.B]([b.sub.1],[b.sub.2]) [less than or equal to] [absolute value of h(a,[b.sub.1]) - h(a,[b.sub.2])] + [epsilon]/3. (5)

Then we choose z [member of] Y such that

[d.sub.A](z,a) < [epsilon]/3. (6)

We have that [absolute value of h(a, [b.sub.1]) - h(a, [b.sub.2])] is bounded above by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This, combined with (5) and (6), completes the proof of (4).

We shall now assume that (B, [d.sub.B]) is not totally bounded and show that this leads to a contradiction. By this assumption and by Proposition 2.5, there exists some positive number r and some infinite sequence [{[b.sub.n]}.sub.n[member of]N] of elements of B such that

[d.sub.B]([b.sub.m],[b.sub.n]) > r for each m,n [member of] N with m [not equal to] n. (7)

For each fixed y [member of] Y it follows from (2) that the numerical sequence [{h(y, [b.sub.n])}.sub.n[member of]N] is bounded and thus has a convergent subsequence. Since Y is countable we can apply a standard Cantor "diagonalization" argument to obtain a subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [{[b.sub.n]}.sub.n[member of]N] such that [lim.sub.n[right arrow][infinity]]h(y, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]) exists for each y [member of] Y. Therefore, after simply changing our notation, we can assume the existence of an infinite sequence [{[b.sub.n]}.sub.n[member of]N] in B which satisfies (7) and also

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

In view of (4) and (7), for each pair of integers m and n with 0 < m < n there exists an element [y.sub.m,n] [member of] Y such that [absolute value of [absolute value of h([y.sub.m,n], [b.sub.m]) - h([y.sub.m,n], [b.sub.n])] > r, and so, in particular,

[absolute value of h([y.sub.m,m+1], [b.sub.m]) - h([y.sub.m,m+1], [b.sub.m+1])] > r for all m [member of] N. (9)

Our assumption that (A, [d.sub.A]) is totally bounded ensures, by Proposition 2.6, that there exists a strictly increasing sequence of positive integers [{[m.sub.k]}.sub.k[member of]N] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is a Cauchy sequence in (A, [d.sub.A]). Now we set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for each k. We choose some sufficiently large integer N for which

[d.sub.A] ([z.sub.N] , [z.sub.k]) < r/4 for all k [greater than or equal to] N. (10)

Now we combine (9) and (10) to obtain that, for each k [greater than or equal to] N,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In view of (8), we obtain that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. So the inequalities on the preceding lines would imply that r [less than or equal to] r/2. This contradiction shows that (B, [d.sub.B]) must be totally bounded, and so completes the proof of the theorem. [there exists]

3. PRELIMINARIES ABOUT LATTICES AND LATTICE COUPLES

Let ([OMEGA], [SIGMA], [mu]) be a [sigma]-finite measure space here and in the sequel. (Some of the assertions which we will be making here are simply false if ([OMEGA], [SIGMA], [mu]) is not [sigma]-finite.)

Definition 3.1. We say that a Banach space X is a CBL, or a complexified Banach lattice of measurable functions on [OMEGA] if

(i) all the elements ofX are (equivalence classes of a.e. equal) measurable functions f : Q [right arrow] C and

(ii) for any measurable functions f : [OMEGA] [right arrow] C and g : [OMEGA] [right arrow] C, if f [member of] X and [absolute value of g] [less than or equal to] [absolute value of f] a.e., then g [member of] X and [[parallel]g[parallel].sub.X] [less than or equal to] [less than or equal to] [[parallel]g[parallel].sub.X].

We will now recall a number of definitions and basic facts about CBLs. In several cases the relevant proofs of these facts in the literature to which we refer are given for Banach lattices of real valued functions. But in all those cases it is an obvious and easy exercise to adapt those proofs to our case here.

Any two CBLs [X.sub.0] and [X.sub.1] on the same underlying measure space always form a Banach couple. See e.g., [3, p. 122 and p. 161], [15, Corollary 1, p. 42], or [12, Remark 1.41, pp. 34-35]. (As explicitly stated and shown in [12] this is also true for non [sigma]-finite measure spaces.)

For each CBL X on ([OMEGA], [SIGMA], [mu]), there exists a measurable subset [[OMEGA].sub.X] of [OMEGA], which may be called the support of X, such that, for every function g [member of] X, we have g([omega]) = 0 for a.e. [omega] [member of] [OMEGA]\[[OMEGA].sub.X]. Furthermore, there exists a function [f.sub.X] [member of] X such that [f.sub.X]([omega]) > 0 for a.e. [omega] [member of] [[OMEGA].sub.X]. (Cf. e.g., Remarks 1.3 and 1.4 on p. 14 of [12].) Obviously the set [[OMEGA].sub.X] is unique to within a set of measure zero. (Of course, on the other hand, the function [f.sub.X] certainly is not unique.) If [[OMEGA].sub.X] = [OMEGA] (at least to within a set of measure zero) then we say that X is saturated.

The set [[OMEGA].sub.X] has an additional useful property: There exists a sequence of sets [{[E.sub.n]}.sub.n[member of]N] in [SIGMA] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The actual construction of [[OMEGA].sub.X] and of the sequence [{[E.sub.n]}.sub.n[member of]N] can be performed by an "exhaustion" process described in the proof of Theorem 3 on pp. 455-456 of [24] and also described (perhaps slightly more explicitly for our purposes here) in the first part of the proof of Proposition 4.1 on p. 58 of [11]. (Note however that there is a small misprint in [11], the omission of "[mu](E)", in the third line of this latter proof, i.e., the numbers [[alpha].sub.k] must of course be defined by [[alpha].sub.k] = sup {[mu](E) : E [member of] [SIGMA], E [subset] [F.sub.k], [[chi].sub.E] [member of] X}.) For one possible (very easy and of course not unique) way to construct a function [f.sub.X] [member of] X with the above-mentioned property see, e.g., [12, p. 14 Remark 1.4].

Lemma 3.2. If [X.sub.0] and [X.sub.1] are both saturated CBLs on the same measure space ([OMEGA], [SIGMA],[mu]), then [X.sub.0] [intersection] [X.sub.1] is saturated, and [[[X.sub.0], [X.sub.1]].sub.[theta]] is saturated for each [theta] [member of] (0,1).

Proof. The function min{[fX.sub.0], [fX.sub.1]} is in [X.sub.0] [intersection] [X.sub.1] and therefore it is also in [[[X.sub.0],[X.sub.1]].sub.[theta]]. It is strictly positive a.e. on [OMEGA]. So neither of the sets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] can have positive measure. [there exists]

Given an arbitrary CBLX on ([OMEGA], [SIGMA], [mu]) we define the functional [[parallel]*[parallel].sub.X'] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

for each measurable function f : [OMEGA] [right arrow] C.

Remark 3.3. Obviously we can replace [absolute value of [[integral].sub.[OMEGA]] fgd[mu]] by [absolute value of [[integral].sub.[OMEGA]][absolute value of fg]d[mu] in the formula (12).

Let X' be the set of all measurable functions f : [OMEGA] [right arrow] C for which [[parallel]f[parallel].sub.X'], < [infinity]. Clearly X' is a linear space and [[parallel]*[parallel].sub.X'] is a seminorm on X' satisfying

[absolute value of [[integral].sub.[OMEGA]] fgd[mu]] [less than or equal to] [[parallel]f[parallel].sub.X'] [[parallel]g[parallel].sub.x] for all f [member of] X' and all g [member of] X. (13)

The space X' is customarily referred to as the Kothe dual or the associate space of X.

If [mu]([[OMEGA].sub.X]) > 0, then, via a series of theorems, including one (Theorem 1, p. 470 in [24]) which uses Hilbert space techniques, it can be shown that X' is non-trivial, i.e., it contains elements which do not vanish a.e. on [[OMEGA].sub.X]. If, furthermore, X is saturated, then [[parallel]*[parallel].sub.X'] is a norm with respect to whichX' is a saturated CBL on ([OMEGA], [SIGMA], [mu]). (See e.g., [24, p. 472, Theorem 4].)

Of course X' can be identified with a subspace of [X.sup.*], the dual space of X, and in some, but not all, cases it is also a norming subspace of [X.sup.*], i.e., it satisfies

[[parallel]g[parallel]].sub.X] = sup {[absolute value of [[integral].sub.[OMEGA]] fgd[mu]] : f [member of] X', [[parallel]f[parallel].sub.x],[less than or equal to] 1}for each g [member of] X. (14)

A result of Lorentz and Luxemburg, which appears as Proposition 1.b.18 on p. 29 of [17] (stated and proved there only for Kothe function spaces), gives necessary and sufficient conditions on X for (14) to hold. In particular the [sigma]-order continuity of X is a sufficient condition. So is the Fatou property. (To extend the proof of Proposition 1.b.18 to general CBLs it is necessary to make a small modification on lines 6 and 7 on p. 30 using Remark 1.3 on p. 14 of [12].)

The associate space (X')' of X', i.e. the second associate of X, is usually denoted by X". Obviously X C X" and [[parallel]x[parallel].sub.X"] [less than or equal to] [[parallel]x[parallel].sub.X] for each x [member of] X. Obviously X" is a CBL whenever X (and therefore also X') is saturated.

As in e.g., [15], we say that the CBL X has absolutely continuous norm if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for every f [member of] X and every sequence [{[E.sub.n]}.sub.n[member of]N] of measurable sets satisfying [E.sub.n+1] [subset] [E.sub.n] for all n and [[intersection].sub.n[member of]N][E.sub.n] = 0.

As in e.g., [17], we say that the CBL X is [sigma]-order continuous if [lim.sub.n[right arrow][infinity]] [[parallel][f.sub.n][parallel].sub.X] = 0 for every sequence [{[f.sub.n]}.sub.n[member of]N] of functions in X satisfying 0 [less than or equal to] [f.sub.n+1] [less than or equal to] [f.sub.n] and [lim.sub.n[right arrow][infinity]] [f.sub.n] = 0 a.e. It is easy to see that these two properties of X are in fact equivalent.

A CBL X is said to have the Fatou property if whenever [{[f.sub.n]}.sub.n[member of]N] is a norm bounded a.e. monotonically non-decreasing sequence of nonnegative functions in X, its a.e. pointwise limit f is also in X with [[parallel]f[parallel].sub.X] = [lim.sub.n[right arrow][infinity]] [[parallel][f.sub.n][parallel].sub.X]. If X is saturated, then X has the Fatou property if and only if X = X" isometrically. (See [24, p. 472]. Cf. also [17, p. 30], but recall that there extra hypotheses are imposed.)

We remark that obvious counterexamples (see e.g., [12, Remark 7.3 p. 92]) show that the above claims about X' and X" are false for certain non [sigma]-finite measure spaces.

Given a pair of CBLs [X.sub.0] and [X.sub.1] on ([OMEGA], [SIGMA], [mu]) and a number [theta] [member of] (0,1), we define the space [X.sub.0.sup.1-[theta]][X.sub.1.sup.[theta]], analogously to the definition in [3, Section 13.5 p. 123], to be the set of all measurable functions f : [OMEGA] [right arrow] C of the form

f = [uf.sub.0.sup.1-[theta]][f.sub.1.sup.[theta]], (15)

where u [member of] [L.sup.[infinity]]([mu]) and [f.sub.j] is a nonnegative function in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for j = 0,1. For each f [member of] [X.sub.0.sup.1-[infinity]][X.sub.1.sup.[theta]] we define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], where the infimum is taken over all representations of f of the form (15) with the stated properties. It can be shown that this is in fact a norm on [X.sub.0.sup.1-[infinity]][X.sub.1.sup.[theta]], with respect to which [X.sub.0.sup.1-[infinity]][X.sub.1.sup.[theta]] is a CBL. This is proved in Section 33.5 on pp. 164-165 of [3].

The norm 1 inclusions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)

are special cases (set [B.sub.0] = [B.sub.1] = C) of the results (i) and (ii) of Section 13.6 on p. 125 of [3] (proved in [3, Section 33.6 on pp. 171-180]). Furthermore, with the help of Bergh's theorem [2], (16) can be strengthened to tell us that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (17)

We will need to use the formula

[([X.sub.0.sup.1-[infinity]][X.sub.1.sup.[theta]])' = [0([X'.sub.0]).sup.1-[theta]][([X'.sub.1]).sup.[theta]], (18)

which holds with equality of norms (or seminorms when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is strictly smaller than [OMEGA]) for all pairs of CBLs [X.sub.0] and [X.sub.1] on ([OMEGA], [SIGMA], [mu]). This formula was originally stated and proved by Lozanovskii [18] under certain hypotheses, then by Reisner [22] under other hypotheses. The general version stated here is proved in [12, Section 7, pp. 91-97] using Reisner's proof and a remark of N. J. Kalton [pers. comm.].

4. THE MAIN RESULT

Our main result is a corollary of the following theorem.

Theorem 4.1. Let [??] = ([G.sub.0], [G.sub.1]) be an arbitrary Banach couple and let [??] = ([X.sub.0],[X.sub.1]) be an arbitrary couple of saturated CBLs on an arbitrary [sigma]-finite measure space ([OMEGA], [SIGMA], [mu]). Then every linear operator T which satisfies T : [??] [??] [??] has the compactness property

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for each [theta] [member of] (0, 1 ).

Corollary 4.2. Let [??] = ([X.sub.0], [X.sub.1]) be an arbitrary couple of saturated CBLs on a [sigma]-finite measure space. Suppose that either

(i) [X.sub.0] and [X.sub.1] both have the Fatou property, or

(ii) at least one ofthe spaces [X.sub.0] and [X.sub.1] is [sigma]-order continuous. Then

(*,*) * [??].

Remark 4.3. The requirement that [X.sub.0] and [X.sub.1] are both saturated is merely a technical convenience which makes the formulation and proof of Theorem 4.1 simpler and shorter. In fact, it is entirely unnecessary for Corollary 4.2. The easy and rather obvious extension of the proof of Corollary 4.2 to the nonsaturated case uses the easily checked fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and replaces the spaces [X.sub.0] and [X.sub.1] in an appropriate way by their "restrictions" to the smaller measure space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Via an examination of the proofs of Theorem 4.1 and Corollary 4.2 it is clear that other conditions on the couple X, weaker than those stated in Corollary 4.2, are also sufficient to ensure that (*, *) * [??].

Proof of Theorem 4.1. Since [[[G.sub.0], [G.sub.1]].sub.[theta]] = [[[G.sub.0], [G.sub.1]].sub.[theta]] [3, Sections 9.3 (p. 116) and 29.3 (pp. 113-114)] we can clearly suppose without loss of generality that G is a regular couple. Let (?, ?) denote the duality between [G.sub.0] [intersection] [G.sub.1] and [([G.sub.0] [intersection] [G.sub.1]).sup.*]. Let G be any one of the spaces [G.sub.0], [G.sub.1] or [[[G.sub.0], [G.sub.1]].sub.[theta]] and define [G.sup.#] to be the subspace of elements [gamma] [member of] [([G.sub.0] [intersection] [G.sub.1]).sup.*] for which the norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is finite. Of course [G.sup.#], when equipped with this norm, is a Banach space which is continuously embedded in [([G.sub.0] [intersection] [G.sub.1]).sup.*]. So ([G.sup.#.sub.0], [G.sup.#.sub.1]) is a Banach couple.

We could of course identify [G.sup.#] with the dual of G, but it is more convenient to use the above definition. Note also that in fact [G.sup.#.sub.0] + [G.sup.#.sub.1] [??] [([G.sub.0] [intersection] [G.sub.1]).sup.*]. Calderon's remarkable duality theorem [3, Section 12.1 p. 121 and Section 32.1 pp. 148-156] can be expressed by the formula [([[G.sub.0],[G.sub.1]].sub.[thata]]).sup.#] [??] [[[G.sup.#.sub.0],[G.sup.#.sub.1]].sup.[theta]]. For a more detailed discussion of all these issues we refer to [6].

Let T be an arbitrary linear operator satisfying [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. We may suppose, without loss of generality, that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. For j = 0 ,1, let [X'.sub.j] be the associate space of [X.sub.j]. For each g [member of] [G.sub.0] [intersection] [G.sub.1] and each z [member of] [X'.sub.0] + [X'.sub.1] define h(g, z) = [[integral].sub.[OMEGA]] zTgd[mu]. Of course (cf. (13)) the function h satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)

for j = 0,1 and all g [member of] [G.sub.0] [intersection] [G.sub.1] and z [member of] [X'.sub.j]. Therefore h also satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (20)

For each fixed z E [X'.sub.0] + [X'.sub.1] we define the linear functional Sz on [G.sub.0] [intersection] [G.sub.1] by (g, Sz) = h(g, z). Of course Sz depends linearly on z and it is clear from (20) that we have thus defined a bounded linear operator S : [X'.sub.0] + [X'.sub.1] [right arrow] [([G.sub.0] [intersection] [G.sub.1]).sup.*]. For j = 0,1, in view of (19), we see that, for each z [member of] [X'.sub.j], we have Sz [member of] [G.sup.#.sub.j] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (Note, cf. [6], that we do not have to consider the extension of Sz to a space larger than [G.sub.0] [intersection] [G.sub.1].)

We now wish to show that S satisfies the compactness condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (21)

We will do this by applying Theorem 2.7. We consider the restriction ofthe function h(g, y) to the set A x B where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Given any sequence [{[g.sub.n]}.sub.n[member of]N] in A, we ofcourse have (cf. (13) and (19)) that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. So the fact that T : [G.sub.0] [??] [X.sub.0] implies that (A,[d.sub.A]) is totally bounded. (Cf., e.g., Proposition 2.5 or Theorem 15 on p. 22 of [13].) Consequently, in view of Theorem 2.7 and Proposition 2.6, if [{[z.sub.N]}.sub.n[member of]N] is an arbitrary sequence in B, then it has a subsequence which is Cauchy with respect to the semimetric

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This is exactly the condition (21).

Since [X'.sub.0] and [X'.sub.1] are both CBLs of measurable functions on the measure space ([OMEGA], [SIGMA], [mu]), we can use (21) and S : [X'.sub.1] [??] [G.sup.#.sub.1] and apply part (c) of Corollary 7 on p. 270 of [8] to deduce that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (22)

We are now ready for a second application of Theorem 2.7. Once more we will use the same function h defined above and restricted to a set A x B, where this time we choose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. This time, for each y,z [member of] B, we of course have S(y - z) [member of] [[[G.sup.#.sub.0], [G.sup.#.sub.1]].sub.[theta]]. So, using the isometry [([[[G.sub.0],[G.sub.1]].sub.[theta]]).sup.#] [??] [[G.sup.#.sub.0], [G.sup.#.sub.1]].sup[theta]] mentioned above, and then Bergh's theorem [2], we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The compactness property (22) of S implies that (B, [d.sub.B]) is totally bounded. Consequently, by Theorem 2.7, (A, [d.sub.A]) is also totally bounded. In view of Proposition 2.6 and the fact that [G.sub.0] [intersection] [G.sub.1] is dense in [[[G.sub.0], [G.sub.1]].sub.[theta]] [3, Section 9.3 (p. 116) and Section 29.3 (pp. 113-114)], this means that the proof of Theorem 4.1 will be complete once we have shown that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (23)

By definition, for each [g.sub.1] and [g.sub.2] in A we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

At this stage we need not consider the particular form of the element [Tg.sub.1] - [Tg.sub.2]. We know that it is an element of [X.sub.0] [intersection] [X.sub.1]. So, to obtain (23) it suffices to show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (24)

Since [X.sub.0] [intersection] [X.sub.1] [subset] [X".sub.0] [intersection] [X".sub.1] [subset] [[X".sub.0] [intersection] [X".sub.1].sub.[theta]], we have, from (17) applied to the couple ([X".sub.0], [X".sub.1]), that the right side of (24) equals [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and this in turn, in view of (18) applied to the couple ([X'.sub.0], [X'.sub.1]), equals [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. It follows that (24) is equivalent to the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (25)

Applying (17) to the couple ([X'.sub.0], [X'.sub.1]), we of course obtain the inequality "[less than or equal to]" in (25). To show the reverse inequality "[greater than or equal to]", we fix some x [member of] [X.sub.0] [intersection] [X.sub.1] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and we shall construct a sequence [{[y.sub.n]}.sub.n[member of]N] in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (26)

By Lemma 3.2, since [X'.sub.0] and [X'.sub.1] are both saturated, so is [[[X'.sub.0],[X'.sub.1]].sub.[theta]]. Consequently (cf. (11)) there exists an expanding sequence [{[E.sub.n]}.sub.n[member of]N] of sets in [SIGMA] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for each n [member of] N. Let [y.sub.n] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Then [y.sub.n] [member of] [[[X'.sub.0],[X'.sub.1]].sub.[theta]] and we have [absolute value of [xy.sub.n]] [less than or equal to] [absolute value of xy] and [lim.sub.n[right arrow][infinity]]x([omega])[y.sub.n]([omega]) = x([omega])y([omega]) for all [omega] [member of] [OMEGA]. The function xy is integrable, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [([X'.sub.0]).sup.1-[theta]][([X'.sub.1]).sup.[theta]] [??] ([X.sup.1-[theta].sub.0][X.sup.[theta].sub.1])' (cf. (17) and (18) and Remark 3.3). So (26) follows from the Lebesgue dominated convergence theorem. As already explained, this implies (25) and therefore also (24) and (23), and so completes the proof of the theorem. [there exists]

Proof of Corollary 4.2. This uses all steps of the preceding proof up to (23). Then it is required to establish a variant of (23) or of (24) where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is replaced by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] or (cf. (17)) by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. If [X.sub.0] and [X.sub.1] both have the Fatou property then [X".sub.0] = [X.sub.0] and [X".sub.1] = [X.sub.1] and we are done. Otherwise, in view of (25) and (18), it will suffice if we show that sup [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for each x [member of] [X.sub.0] [intersection] [X.sub.1].

This will hold whenever ([X.sup.1-[theta]][X.sup.[theta].sub.1])' is a norming subspace of the dual of [X.sup.1-[theta]][X.sup.[theta].sub.1] (and possibly also under a weaker assumption than that, since we are only considering elements x in [X.sub.0] [intersection] [X.sub.1]). As already observed above (just after (14)), one sufficient condition for this to happen is when [X.sup.1-[theta]][X.sup.[theta].sub.1] is [sigma]-order continuous. The [sigma]-order continuity of [X.sub.0.sup.1-[theta]][X.sub.1.sup.[theta]] can be ensured by requiring that at least one of the spaces [X.sub.0] and [X.sub.1] is [sigma]-order continuous (cf. Proposition 4 on p. 80 of [22] or Theorem 1.29 on p. 27 of [12]). [there exists]

ACKNOWLEDGEMENTS

The research was supported by the Technion V.P.R. Fund and by the Fund for Promotion of Research at the Technion.

Received 5 October 2009, accepted 2 February 2010

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Michael Cwikel

Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel; mcwikel@math.technion.ac.il
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