INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES


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1 INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the representing polymeasure of the operator. Some applications are given. In particular, we characterize the Borel polymeasures that can be extended to a measure in the product σ algebra, generalizing previous results for bimeasures. We also give necessary conditions for the wea compactness of the extension of an integral multilinear operator on a product of C(K) spaces. 1. Introduction The modern theory of Banach spaces is greatly indebted to the wor of A. Grothendiec. In his papers [11] and [12] he introduced the most important classes of operator ideals, whose study and characterization in different concrete classes of Banach spaces has been a permanent subject of interest since then. One of the classes defined in [11] and now intensively studied, is the class of integral operators (see below), whose definition establishes a first connection between the linear and the multilinear (bilinear, in fact) theory. Grothendiec himself started the study of several classes of operators on C(K) spaces in [12]. As a consequence of the Riesz representation Theorem, every continuous linear map T from C(K) into another Banach space X has a representing measure, i.e., a finitely additive measure m of bounded semi variation defined on the Borel σfield of K, with values in X (the bidual of X), in such a way that T (f) := f dm, for each f C(K). (see, e.g. [5] or [6]). The study of the relationships between T and its representing measure plays a central role in this research. When T is a continuous linear map from a product C(K 1 ) C(K ) (where K i are compact Hausdorff spaces) into a Banach space X, there exists also an integral representation theorem with respect to the representing polymeasure of T (see below for the definitions). If = 1, the integral operators (Gintegral in our notation; see Definition 2.3 below) are precisely those whose representing measure has bounded variation (see e.g. [16, p. 477] and [5, Th. VI.3.3]). The aim of this paper is to study and characterize the multilinear vector valued integral operators on a product of C(K) spaces in terms of the corresponding representing polymeasure. As an application we obtain an intrinsic characterization of the Borel polymeasures than can be extended to measures in the product Borel σalgebra, extending some previous results for the case of bimeasures. We also study the relationship between the wea compactness of an integral multilinear map on a Both authors were partially supported by DGICYT grant PB
2 2 FERNANDO BOMBAL AND IGNACIO VILLANUEVA product C(K 1 ) C(K ) and that of its linear extension to C(K 1 K ). Some other applications are given. 2. Definitions and Preliminaries The notation and terminology used throughout the paper will be the standard in Banach space theory, as for instance in [5]. However, before going any further, we shall establish some terminology: L (E 1..., E ; X) will be the Banach space of all the continuous linear mappings from E 1 E into X and L wc(e 1..., E ; X) will be the closed subspace of it formed by the wealy compact multilinear operators. When X = K or = 1, we will omit them. If T L (E 1..., E ; X) we shall denote by ˆT : E 1 E X its linearization. As usual, E 1 ˆ ɛ ˆ ɛ E will stand for the (complete) injective tensor product of the Banach spaces E 1,..., E. We shall use the convention. [i].. to mean that the ith coordinate is not involved. If T L (E 1,..., E ; X) we denote by T i (1 i ) the operator T i L(E i ; L 1 (E 1,. [i].., E ; X) defined by T i (x i )(x 1,. [i].., x ) := T (x 1,..., x ), Let now Σ i (1 i ) be σalgebras (or simply algebras) of subsets on some non void sets Ω i. A function γ : Σ 1 Σ X or γ : Σ 1 Σ [0, + ] is a (countably additive) polymeasure if it is separately (countably) additive. ([8, Definition 1]). A countably additive polymeasure γ is uniform in the i th variable if the measures {γ(a 1,..., A i 1,, A i+1,..., A ) : A j Σ j (j i)} are uniformly countably additive. As in the case = 1 we can define the variation of a polymeasure γ : Σ 1 Σ X, as the set function given by v(γ) : Σ 1 Σ [0, + ] n 1 v(γ)(a 1,..., A ) = sup =1 n j =1 γ(a 1,... A j ) where the supremum is taen over all the finite Σ i partitions (A j i i )n i j i =1 of A i (1 i ). We will call bvpm(σ 1,..., Σ ; X) the Banach space of the polymeasures with bounded variation defined on Σ 1 Σ with values in X, endowed with the variation norm. We can define also its semivariation by γ : Σ 1 Σ [0, + ] n 1 γ (A 1,..., A ) = sup =1 n j =1 a 1... a j γ(a 1,..., A j ) where the supremun is taen over all the finite Σ i partitions (A j i i )n i j i =1 of A i (1 i ), and all the collections (a ji i )ni j i =1 contained in the unit ball of the scalar field. We will call bpm(σ 1,..., Σ ; X) the Banach space of the polymeasures with bounded semivariation defined on Σ 1 Σ with values in X, endowed with the semivariation norm.
3 INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES 3 If γ has finite semivariation, an elementary integral (f 1, f 2,... f ) dγ can be defined, where f i are bounded, Σ i measurable scalar functions, just taing the limit of the integrals of uples of simple functions (with the obvious definition) uniformly converging to the f i s (see [8]). If K 1,..., K are compact Hausdorff spaces, then every multilinear operator T L (C(K 1 ),..., C(K ); X) has a unique representing polymeasure γ : Bo(K 1 ) Bo(K ) X (where Bo(K) denotes the Borel σalgebra of K) with finite semivariation, in such a way that T (f 1,..., f ) = (f 1,..., f ) dγ for f i C(K i ), and such that for every x X, x γ rcapm(bo(k 1 ),..., Bo(K )), the set of all regular, countably additive scalar polymeasures on Bo(K 1 ) Bo(K ). (Cfr. [2]). Given a polymeasure γ we can consider the set function γ m defined on the semiring of all measurable rectangles A 1 A (A i Σ i ) by γ m (A 1 A ) := γ(a 1,..., A ) It follows, f. i. from [7, Prop. 1.2] that γ m is finitely additive and then it can be uniquely extended to a finitely additive measure on the algebra a(σ 1 Σ ) generated by the measurable rectangles. In general, this finitely additive measure cannot be extended to the σalgebra Σ 1 Σ generated by Σ 1 Σ. But if there is a countably additive measure µ of bounded variation on Σ 1 Σ that extends γ m, then by standard measure theory (see e.g. [6, Th. I.5.3] and [7]) we have v(γ m )(A 1 A ) = v(γ)(a 1,..., A ) = v(µ)(a 1 A ), for A i Σ i. ( ) The next definition extends Grothendiec s notion of multilinear integral forms to the multilinear integral operators: Definition 2.1. A multilinear operator T L (E 1,..., E ; X) is integral if ˆT ( i.e., its linearization) is continuous for the injective (ɛ) topology on E 1 E. Its norm (as an element of L(E 1 ˆ ɛ ˆ ɛ E ; X)) is the integral norm of T, T int := ˆT ɛ. Proposition 2.2. T L (E 1,..., E ; X) is integral if and only if x T is integral for every x X. Proof. For the nontrivial part, let us consider the map X x x ˆT (E 1 ˆ ɛ ˆ ɛ E ), well defined by hypothesis. A simple application of the closed graph theorem proves that this linear map is continuous. Hence, sup x ˆT ɛ = M <. x 1 But it is easy to see that T int := sup u ɛ 1 ˆT (u) = M. The next definition is well nown. Definition 2.3. An operator S L(E; X) is Gintegral (the G comes from Grothendiec ) if the associated bilinear form B S : E X K (x, y) y(t (x))
4 4 FERNANDO BOMBAL AND IGNACIO VILLANUEVA is integral. In that case the integral norm of S, S int := B S int. Let us recall that a bilinear form T L 2 (E 1, E 2 ) is integral if and only if any of the two associated linear operators T 1 L(E 1 ; E 2) and T 2 L(E 2 ; E 1) is Gintegral in the above sense (cfr., e.g. [5, Ch VI]). Proposition 2.4. Let 2, E 1,..., E be Banach spaces and T L (E 1,..., E ). Then T is integral if and only if there exists i, 1 i, such that a) For every x i E i, T i (x i ) is integral. b) The mapping [i] T i : E i (E 1 ɛ ɛ E ) defined by T i (x i ) := T i (x i ) is a Gintegral operator. If (a) and (b) are satisfied for some i, then the same happens for any other index j, 1 j. Moreover, in this case, T int = T i int. Proof. If (a) and (b) are satisfied and we put F i := E 1 ɛ ɛ E, the bilinear map B Ti : E i F i K is integral and B Ti int = T i ([5, Corollary VIII.2.12]). From the associativity, the commutativity of the ɛtensor product and the definitions, it follows that T is integral and the norms are equal. Conversely, suppose that T is integral. We shall prove that (a) and (b) hold for i = 1: From the hypothesis and the associativity of the injective tensor product, it follows that the bilinear map B T : E 1 (E 2 ˆ ɛ ˆ ɛ E ) K [i] (x, u) ˆT (x u) is integral. By [5, Corollary VIII.2.12], the associated linear operator from E 1 into (E 2 ˆ ɛ ˆ ɛ E ) is Gintegral. Clearly, this operator coincides with T 1, and this proves (a) and (b) for i = Integral forms on C(K) spaces Let now K, K 1,..., K be compact Hausdorff spaces. Recall that, for every Banach space X, C(K, X), the Banach space of all the Xvalued continuous functions on K endowed with the sup norm, is canonically isometric to C(K) ˆ ɛ X ([5, Example VIII.1.6]). Moreover, if X = C(S) (S a compact Hausdorff space) then C(K, C(S)) is canonically isometric to C(K S). Thus, we have the following identifications C(K 1 ) ˆ ɛ ˆ ɛ C(K ) C(K i, C(K 1 ) ˆ ɛ [i] ˆ ɛ C(K ) C(K 1 K ). Suppose that T L (C(K 1 ),..., C(K )) with representing polymeasure γ. If there exists a regular measure µ on the Borel σalgebra of K 1 K that extends γ m, then, by the Riesz representation theorem, µ is the representing measure of some continuous linear form ˆT on C(K 1 K ) C(K 1 ) ˆ ɛ ˆ ɛ C(K ) and clearly (1) ˆT (f 1 f ) = f 1 f dµ = K 1 K = T (f 1,..., f ) = (f 1,..., f )dγ. K 1 K
5 INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES 5 Consequently T is integral. Note also that, as follows from the introduction, ˆT = v(µ) = v(γ). Conversely, if T is such that its linearization ˆT on C(K 1 ) C(K ) is continuous for the ɛtopology (i.e., T is integral), another application of the Riesz representation theorem yields a measure µ on Bo(K 1 K ) such that (1) holds. By the uniqueness of the representation theorem for linear maps, we have µ(a 1 A ) = γ(a 1,..., A ) (for every A i Σ i ) and so µ extends γ. Summarizing, we have proved Proposition 3.1. Let 2 and T L (C(K 1 ),..., C(K )) with representing polymeasure γ. Then T is integral if and only if γ can be extended to a regular measure µ on Bo(K 1 K ) in such a way that µ(a 1 A ) = γ(a 1,..., A ) In this case, ˆT = T int = v(γ) = v(µ). for every A i Σ i, (1 i ). Consequently, (C(K 1 ) ˆ ɛ ˆ ɛ C(K )) can be isometrically identified with a subspace of the space of all regular polymeasures on Bo(K 1 ) Bo(K ) with finite variation, endowed with the variation norm. Now we are going to obtain an intrinsic characterization of the extendible Radon polymeasures which will allow us to see that the previous isometry is onto. If Σ 1,..., Σ are σalgebras, X is a Banach space and γ bpm(σ 1,..., Σ ; X), then we can define a measure by ϕ 1 : Σ 1 bpm(σ 2,..., Σ ; X) ϕ 1 (A 1 )(A 2,..., A ) = γ(a 1, A 2,..., A ). It is nown that ϕ 1 = γ (see [3]). Related to this we have the following lemma, whose easy proof we include for completeness: Lemma 3.2. Let X be a Banach space, Ω 1,..., Ω sets and Σ 1,..., Σ σalgebras defined on them. Let now γ : Σ 1 Σ X be a polymeasure. Then v(γ) < if and only if ϕ 1 taes values in bvpm(σ 2,... Σ ; X) and v(ϕ 1 ) < when we consider the variation norm in the image space. In that case, v(ϕ 1 )(A 1 ) = v(γ)(a 1, Ω 2,..., Ω ) and v(ϕ 1 (A 1 ))(A 2,... A ) v(γ)(a 1, A 2,..., A ). Of course the role played by the first variable could be played by any of the other variables. Proof. Let us first assume that v(γ) <. In the following we will adopt the convention that sup j2...,j means the supremum over all the finite Σ i partitions (A ji i )ni j i =1 of A i (2 i ). Then, with this notation, v(ϕ 1 (A 1 )) = v(ϕ 1 (A 1 ))(Ω 2,..., Ω ) = (2) = sup j 2...,j j 2 j sup j 2,...,j j 2 j ϕ 1 (A 1 )(A i2 2,..., A i ) = γ(a 1, A j2 2,..., A j ) v(γ)(a 1, Ω 2,..., Ω ) v(γ)(ω 1, Ω 2,..., Ω ) = v(γ). Therefore, ϕ 1 is bvpm(σ 2,..., Σ ; X)valued. Let us now see that it has bounded variation when we consider the variation norm in the image space:
6 6 FERNANDO BOMBAL AND IGNACIO VILLANUEVA (3) v(ϕ 1 ) = v(ϕ 1 )(Ω 1 ) = sup sup ϕ 1 (A j1 1 ) = sup v(ϕ 1 (A j1 1 )) v(γ)(a j1 1, Ω 2,..., Ω ) v(γ)(ω 1, Ω 2,..., Ω ) = v(γ) <. In the next to last inequality we have used that the variation of a polymeasure is itself separately countably additive ([8, Theorem 3]). Conversely, if ϕ 1 is bvpm(σ 2,..., Σ ; X)valued and with bounded variation when we consider the variation norm in the image space, then (4) v(γ) = v(γ)(ω 1,..., Ω ) = sup γ(a j1 1, A j 2 2,..., A j ),j 2,...,j j 2 j sup sup ϕ 1 (A j1 1 )(A j2 2,..., Aj ) = sup ϕ 1 (A j1 1 ) = j j 2,...,j j 2 = v(ϕ 1 )(Ω 1 ) = v(ϕ 1 ) <. Putting together both inequalities we get that v(γ) = v(ϕ 1 ). To prove the first of the last two statements of the lemma we replace Ω 1 by A 1 in (3), (4). To prove the last statement we replace (Ω 2,...,Ω ) by (A 2,..., A ) in (2). Let T L (C(K 1 ),..., C(K )) with representing polymeasure γ. Let us consider T 1 : C(K 1 ) L 1 (C(K 2 ),..., C(K )) and let ϕ 1 : Bo(K 1 ) rcapm(bo(k 2 ),..., Bo(K )) be defined as above. It is nown that ϕ 1 is countably additive if and only if γ is uniform ([3, Lemma 2.2]) and in this case ϕ 1 is the representing measure of T 1 ([3, Theorem 2.4]). From the definitions, it is easy to chec that every polymeasure with finite variation is uniform. Now we can prove the first of our main results. Theorem 3.3. Let T L (C(K 1 ),..., C(K )) with representing polymeasure γ. Then the following are equivalent: a) v(γ) <. b) T is integral. c) γ can be extended to a regular measure µ on Bo(K 1 K ). d) γ can be extended to a countably additive (not necessarily regular) measure µ 2 on Bo(K 1 ) Bo(K ). Proof. The equivalence between (b) and (c) is just Proposition 3.1. If (c) holds, defining µ 2 := µ Bo(K1 ) Bo(K ) proves (d). Since a countably additive scalar measure has bounded variation, from ( ) in the previous Section, we get that (d) implies (a). Finally let us prove that (a) implies (b): By Proposition 2.4 we have to show that i) T 1 (f 1 ) (C(K 2 ) ˆ ɛ ˆ ɛ C(K )) := F and ii) T 1 : C(K 1 ) F is Gintegral.
7 INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES 7 We shall proceed by induction on. For = 1 there is nothing to prove. Let = 2. In this case we only have to prove (ii). By the discussion following Lemma 3.2, the representing measure of T 1 is ϕ 1 : Bo(K 1 ) C(K 2 ) and v(ϕ 1 ) = v(γ) <. Since every dual space is 1complemented in its bidual, by Corollary VIII.2.10 and Theorems VI.3.3 and VI.3.12 of [5], T 1 is integral and T 1 int = ˆT = v(ϕ) = v(γ) by Lemma 3.2. Let us now suppose the result true for 1. Let T L (C(K 1 ),..., C(K )). i) For f 1 C(K 1 ), the representing polymeasure of T 1 (f 1 ) is γ f1, defined by γ f1 (A 2,..., A ) = (f 1, χ A2,..., χ A ) dγ, as can be easily checed. Since γ f1 (A 2,..., A ) f 1 v(γ)(k 1, A 2,..., A ), it follows that γ f1 has finite variation and v(γ f1 ) v(γ) f 1. Hence, by the induction hypothesis, T 1 (f 1 ) is integral. ii) As before, we have to prove that the representing measure of T 1 : C(K 1 ) (C(K 2 ) ˆ ɛ ˆ ɛ C(K )) has finite variation. The representing measure of T 1 is ϕ 1, where ϕ 1 (A 1 )(f 2 f ) = (χ A1, f 2,..., f )dγ and, clearly, ϕ 1 is just ϕ 1 of Lemma 3.2 considering the integral (equivalently variation) norm in the image space. Therefore, Lemma 3.2 proves that v( ϕ 1 ) = v(γ) <, and so T 1 is Gintegral. Remar 3.4. The equivalence (a) (d) was proved for bimeasures in [14, Corollary 2.9]. The techniques used in that paper, essentially different from ours, do not seem to extend easily to the case of polymeasures when 3. To the best of our nowledge, it was unnown when a polymeasure could be decomposed as the sum of a positive and a negative polymeasure. It is clear now that, for the polymeasures in rcapm(bo(k 1 ),..., Bo(K )), this happens only in the most trivial case, that is, when γ can be extended to a measure, and then decomposed as such. Corollary 3.5. Given γ rcapm(bo(k 1 ),..., Bo(K )), γ can be decomposed as the sum of a positive and a negative polymeasure if and only if v(γ) <. Proof. If v(γ) <, then γ can be extended to µ as in Theorem 3.3. Let us now decompose this measure µ as the sum of a positive and a negative measure µ = µ p + µ n. Clearly now γ = µ p + µ n, considering µ p and µ n as polymeasures. Conversely, if γ = γ p + γ n, where γ p (resp. γ n ) is a positive (resp. negative) polymeasure, then v(γ) = v(γ)(k 1,..., K ) γ p (K 1,..., K ) γ n (K 1,..., K ) <.
8 8 FERNANDO BOMBAL AND IGNACIO VILLANUEVA 4. Vectorvalued integral maps on C(K) spaces We will use now the results of the preceding section to characterize the vector valued integral operators. First we will need a new definition: Let Ω 1,...,Ω be nonempty sets and Σ 1,...,Σ be σalgebras defined on them. If γ : Σ 1 Σ X is a Banach space valued polymeasure, we can define its quasivariation by γ + : Σ 1 Σ [0, + ] n 1 γ + (A 1,..., A ) = sup =1 n j =1 a j1,...,j γ(a 1,j1,..., A,j ) where (A i,ji ) n i j i=1 is a Σ ipartition of A i (1 i ) and a j1,...,j 1 for all (,..., j ). It is not difficult to see that the quasivariation is separately monotone and subadditive and that, for every (A 1,..., A ) Σ 1 Σ, γ (A 1,..., A ) γ + (A 1,..., A ) v(γ)(a 1,..., A ). It can also be checed that γ + = sup{v(x γ); x B X }. We can consider the space of polymeasures γ bpm(σ 1,..., Σ ; X) such that γ + <. Standard calculations show that + is a Banach space norm in this space. This space has been recently considered in [7], where the authors develop a theory of integration for these polymeasures. We will prove in this section that, for the polymeasures representing multilinear operators on C(K 1 ) C(K ), the ones with finite quasivariation are precisely those which can be extended to a measure on Bo(K 1 K ), and thus the previously mentioned integration theory can be dispensed with. Theorem 4.1. Let 2, T : C(K 1 ) C(K ) X be a multilinear operator and let γ : Bo(K 1 ) Bo(K ) X be its associated polymeasure. Then the following are equivalent: a) γ + <. b) T is integral. c) γ can be extended to a bounded ω regular measure µ : Bo(K 1 K ) X (in such a way that µ(a 1 A ) = γ(a 1,..., A ) for every A i Σ i, (1 i ) ). d) γ can be extended to a bounded ω countably additive (not necessarily regular) measure µ 2 : Bo(K 1 ) Bo(K ) X. Proof. Let us first prove that (a) implies (b): If γ + <, then, for every x X, v(x γ) <. Since x γ is clearly the representing polymeasure of x T, using Theorem 3.3, we obtain that x T is integral for every x X and now we can apply Proposition 2.2 to finish the proof. Let us now prove that (b) implies (c): Let T L (C(K 1 ),..., C(K ); X) be integral, and call T L(C(K 1 K ); X) its extension. If µ : Bo(K 1 K ) X is the representing measure of T, it is clear that µ satisfies (c). Clearly (c) implies (d). Now, if (d) is true, then µ 2 = sup x B X v(x µ) = sup x B X v(x γ) = γ +, and (a) holds.
9 INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES 9 Corollary 4.2. Let T : C(K 1 ) C(K ) X be a multilinear operator with representing polymeasure γ. Then the following statements are equivalent: a) v(γ) <. b) T is integral and its extension T : C(K 1 K ) X is Gintegral. Proof. If T is integral and µ is the representing measure of its extension T then, as we saw in Section 2, v(γ) = v(µ). Hence, the equivalence between (a) and (b) follows from γ + v(γ) and the fact (already used) that a linear operator on a C(K)space is Gintegral if and only if its representing measure has finite variation. The fact that every integral multilinear map T : C(K 1 ) C(K ) X can be extended to a continuous linear map T : C(K 1 ) ˆ ɛ ˆ ɛ C(K ) C(K 1 K ) X has some immediate consequences: Proposition 4.3. Let T : C(K 1 ) C(K ) X be an integral multilinear operator and for each i, 1 i, let (fi n) C(K i) be bounded sequences. a) If at least one of the sequences (fi n) is wealy null and (x n) X is a wealy Cauchy sequence, then lim T (f 1 n,..., f n ), x n = 0 ( ) n b) If all the sequences (fi n) are wealy Cauchy and (x n) is a wealy null sequence in X, then ( ) holds. Proof. From the well nown characterization of the wea topology in C(K)spaces, it follows that the sequence (f n 1 f n ) C(K 1) ˆ ɛ ˆ ɛ C(K ) C(K 1 K ) is, respectively, wealy null (under (a)) or wealy Cauchy (under (b)), and T (f n 1 ),..., f n ), x n = f n 1 f n, T (x n) The result follows from the DunfordPettis property of C(K 1 K ). If = 2, it can be proved that the sequence (f n 1 f n 2 ) is also wealy null or wealy Cauchy, respectively, in the projective tensor product C(K 1 ) ˆ π C(K 2 ) ([4, Lemma 2.1]). Hence, when X has the DunfordPettis Property the above result is true for any continuous bilinear map. Nevertheless, Proposition 4.3 gives a necessary condition for a multilinear map to be integral, and so it provides an easy way to see when a multilinear map is not integral. Example 4.4. Let (r n ) be a bounded, orthonormal sequence (with respect to the usual scalar product) in C([0, 1]) and let T : C([0, 1]) C([0, 1]) l 2 be defined by (( 1 ) ( ))) 1 T (f, g) = fr n (g 0 n n=1 Then T (clearly wealy compact) is not integral. In fact, if g n denotes the function which is equal to 1 at 1 1 n, 0 in [0, 2n ] and [ 3 2n, 1] and linear elsewhere, the sequence (g n ) converges pointwise to 0 and so is wealy null in C([0, 1]). But T (r n, g n ) = e n, the usual l 2 basis, and so T (r n, g n ), e n = 1 for every n. Let us state a definition: given Banach spaces E 1,..., E, X, a multilinear operator T L (E 1,..., E ; X) is called regular if every one if the linear operators T i L(E i ; L 1 (E 1,. [i].., E ; X)) associated to it are wealy compact (see [1] and [10] for some properties of these operators). In case T L (C(K 1 ),..., C(K ); X),
10 10 FERNANDO BOMBAL AND IGNACIO VILLANUEVA it follows from [3] that its representing polymeasure γ is uniform if and only if T is regular. Proposition 4.5. Let T : C(K 1 ) C(K ) X be an integral multilinear operator with representing polymeasure γ, and suppose that its extension T : C(K 1,, K ) X is wealy compact. Then γ is uniform, and therefore T is regular. Proof. Let us prove, for instance, that γ is uniform in the first variable. According to [3, Theorem 2.4], it suffices to prove that the corresponding operator T 1 : C(K 1 ) L 1 (C(K 2 ),... C(K ); X) is wealy compact or, equivalently, it maps wealy null sequences into norm null sequences ([5, Corollary VI.2.17]). Let (f1 n ) C(K 1 ) be a wealy null sequence. We have to prove that T 1 (f1 n ) 0 when n. If not, there would be an ɛ > 0 and a subsequence (denoted in the same way) such that T 1 (f1 n ) > ɛ for every n. Then we could produce fj n C(K j) (2 j ), fj n 1, such that T 1(f1 n )(f2 n,..., f n) = T (f 1 n,..., f n ) > ɛ for every n. But (f1 n f2 n f n) C(K 1,, K ) converges wealy to 0. Hence, from the aforementioned property of wealy compact operators on C(K)spaces, T (f1 n f n) = T (f 1 n,..., f n ) tends to 0 as n tends to, which is a contradiction. Remar 4.6. Note that if X is reflexive, in particular if X = K, then it follows from the above result that integral operators are regular. We do not now if the converse of the Proposition 4.5 is true. In any case, if T : C(K 1 ) C(K ) X is integral and regular and, for instance, we denote by T 1 : C(K 1 ) L 1 (C(K 2 ),..., C(K ); X) the associated linear map, it is easily checed that T (ϕ 1 ) is integral for any ϕ 1 C(K 1 ), and its representing measure taes also values in the space of integral ( 1)linear operators. Thus, it can be considered as a measure m 1 : Σ 1 L(C(K 2 K ); X). Reasoning in a similar way as in [3, Theorem 2.4], we can prove that m 1 coincides with the representing measure of the operator ˆT : C(K 1, C(K 2 K )) X given by the DinculeanuSinger Theorem (see, e.g. [5, p. 182]), and the wea compactness of T is clearly equivalent to that of ˆT. In the linear case, an operator T : C(K) X is wealy compact if and only if its representing measure µ taes values in X, if and only if µ is countably additive. This is not longer true in the multilinear case, where the role of wealy compact operators seems to be played by the so called completely continuous multilinear maps (see [17]). In the case of integral multilinear maps one could conjecture that the wea compactness and the behaviour of the representing polymeasure of T should be analogous to that of the extended linear operator. This is not true, as the following example shows: Example 4.7. Let us consider l = C(βN). Let q : l l 2 be a linear, continuous and onto map ([15, Remar 2.f.12]), and let us tae a bounded sequence (a n ) l such that q(a n ) = e n (the canonical basis of l 2 ) for any n. Suppose a n C. Then (e n a n ) is a basic sequence in l ˆ ɛ l C(βN, l ) ([13, Proposition 3.15]), equivalent to the canonical basis of c o, since n λ i e i a i ɛ = i=1 sup x 1 n λ i x (a i )e i C (λ i ). i=1
11 INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES 11 Moreover, if ϕ n := e n q (e n) l l (l ˆ ɛ l ) we have ϕ n q for all n and so, as ϕ n (a b) 0 when n tends to, it turns out that (ϕ n ) is a wea null sequence. Hence P (u) := n=1 ϕ n(u)e n a n is a continuous projection from l ˆ ɛ l onto the closed subspace (isomorphic to c 0 ) spanned by {e n a n : n N}. Consequently, ˆT : l ˆ ɛ l c 0 defined as ˆT (u) := (ϕ n (u)) c 0, is linear, continuous and onto. In particular, ˆT is not wealy compact. By construction, the corresponding bilinear map T : l l c 0 is integral. Also, since (ϕ n (x y)) n l 2 for x, y l and T (x, y) 2 q x y, it follows that T factors continuously through l 2 and consequently it is wealy compact. In particular, the representing bimeasure γ of T taes values in c 0, ([2, Corollary 2.2]), but the extended measure µ that represents T : C(βN, l ) c 0 does not. The next proposition characterizes when T is wealy compact in terms of the representing polymeasure of T : Proposition 4.8. Let T : C(K 1 ) C(K ) X be an integral multilinear operator with representing polymeasure γ, and let µ be the representing measure of its extension T : C(K 1 K ) X. The following assertions are equivalent: a) T is wealy compact. b) µ taes values in X. c) µ is countably additive. d) γ m (see Section 2) taes values in X and is strongly additive. Proof. The equivalences between (a), (b) and (c) are well nown (see [5, Theorem VI.2.5]), and obviously they imply (d). Finally, since µ is a w countably additive extension of γ m, (d) implies that γ m is wealy countably additive (and strongly additive). The HahnKluvane extension Theorem ([5, Theorem I.5.2]) provides an (unique) Xvalued countably additive extension of γ m to Σ := Bo(K 1 ) Bo(K ), which clearly coincides with µ. Obviously, every function in C(K 1 ) C(K ) is Σmeasurable. Hence, by density, every continuous function on K 1 K is Σmeasurable. Urysohn s lemma proves that every closed, F σ set belongs to Σ, and so is sent by µ to X. A well nown result of Grothendiec ([12, Théorème 6]) proves that µ sends any Borel subset of K 1 K to X. References [1] R. M. Aron and P. Galindo, Wealy compact multilinear mappings, Proc. of the Edinburgh Math. Soc. 40 (1997) [2] F. Bombal and I. Villanueva, Multilinear operators on spaces of continuous functions. Funct. Approx. Comment. Math. XXVI (1998), [3] F. Bombal and I. Villanueva, Regular multilinear operators in spaces of continuous functions. Bull. Austral. Math. Soc., vol. 60 (1999), [4] F. Bombal and I. Villanueva, On the DunfordPettis property of the tensor product of C(K) spaces. To appear in Proc. Amer. Math. Soc.. [5] J. Diestel and J. J. Uhl, Vector Measures. Mathematical Surveys, No. 15. American Math. Soc., Providence, R.I., [6] N. Dinculeanu, Vector Measures. Pergamon Press, [7] N. Dinculeanu and M. Muthiah, Bimeasures in Banach spaces. Preprint. [8] I. Dobraov, On integration in Banach spaces, VIII (polymeasures). Czech. Math. J. 37 (112) (1987), [9] I. Dobraov, Representation of multilinear operators on C 0 (T i ). Czech. Math. J. 39 (114) (1989),
12 12 FERNANDO BOMBAL AND IGNACIO VILLANUEVA [10] M. González and J. M. Gutiérrez, Injective factorization of holomorphic mappings, Proc. Amer. Math. Soc. 127 (1999), [11] A. Grothendiec, Produits tensoriels topologiques et espaces nuclaires. Mem. Amer. Math. Soc. 16 (1955). [12] A. Grothendiec, Sur les applications linaires faiblement compactes d espaces du type C(K). Canad. J. Math. 5 (1963), [13] J. R. Holub, Tensor product bases and tensor diagonals, Trans. Amer. Math. Soc. 151 (1970), [14] S. Karni and E. Merzbach, On the extension of Bimeasures. Journal d Analyse Math. 55 (1990), [15] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, I Springer, Berlin, [16] J. R. Retherford and C. Stegall, Fully Nuclear and Completely Nuclear Operators with applications to L 1 and L spaces. Trans. Amer. Math. Soc. 163 (1972), [17] I. Villanueva, Completely continuous multilinear operators on C(K) spaces, Proc. of the Amer. Math. Soc. 128 (1999), Departamento de Anlisis Matemtico, Facultad de Matemticas, Universidad Complutense de Madrid, Madrid address: ignacio
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