Characterizations of Banach Spaces With Relatively Compact Dunford-Pettis Sets

Æ Å ADVANCES IN MATHEMATICS(CHINA) doi: 10.11845/sxjz.2014095b Characterizations of Banach Spaces With Relatively Compact Dunford-Pettis Sets WEN Yongming, CHEN Jinxi (School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan, 611731, P. R. China) Abstract: In this paper, we introduce the definition of Dunford-Pettis completely continuous operator. We give some properties of this concept related to some well-known classes of operators, and especially, related to the relatively compact Dunford-Pettis property (DPrcP) of a space X or Y. Then some sufficient conditions for the DPrcP of some Banach spaces, with respect to Dunford-Pettis completely continuity, are verified. Finally, we investigate the dominated property of the Dunford-Pettis completely continuous operators on Banach lattices. Keywords: Dunford-Pettis set; Dunford-Pettis completely continuous operator; DPrcP; evaluation operator; domination MR(2010) Subject Classification: 47L05; 47L20; 46B28 / CLC number: O177 Document code: A 0 Introduction A subset A of a Banach space X is called a Dunford-Pettis (resp. limited) set [2 4], if any weakly null (weak null) sequence (x n) in X converges uniformly on A. That is, lim sup n + x A x,x n = 0. A norm bounded subset A of a Banach space X is a Dunford-Pettis (DP) set whenever every weakly compact operator from X to an arbitrary Banach space carries A to a norm totally bounded set (relatively compact set) [1, p. 350]. This note is devoted to a study of the family of Banach spaces with the property that their DP sets are relatively compact. We shall say that such spaces have the DPrcP in short. A dual Banach space with the weak Radon-Nikodym property (see [16], in short WRNP) has the DPrcP [9]. If a Banach space has the DPrcP, then it has the compact range property (see [16], in short CRP). Moreover, if it is a dual space, then it has the WRNP iff it has the DPrcP [10]. Schur spaces have the DPrcP. By [3, Corollary 3.10], every discrete KB-space has the DPrcP. In [18], Salimi and Moshtaghioun characterized the Gelfand-Phillips (GP) property of Banach spaces by introducing the definition of the limited completely continuous operator. Inspired by this, we introduce the definition of Dunford-Pettis completely continuous (DPcc) operator in Received date: 2014-06-13. Revised date: 2014-10-10. Foundation item: The work is supported by NSFC (No. 11301285). E-mail: lanyang2014@126.com

2 order to characterize the DPrcP. We study the properties of this kind of operators and investigate the DPrcP of some operator spaces. Finally, we survey the domination of positive DPcc operators on Banach lattices. In this paper, the following definitions are often used. (1) If every limited set of a Banach space X is relatively compact, then X is said to have the GP property. If every DP set in X is relatively compact, then X is said to have the relatively compact DP property [10]. It is clear that the DPrcP implies the GP property, but if X is a Grothendieck space (i.e., weak and weak convergence of sequences in X coincide), then these properties are the same. The Banach space X has the GP property if and only if every limited and weakly null sequence (x n ) in X is norm null [8]. Analagously, in this paper, we give an analogous conclusion about the DPrcP. Let X and Y be arbitrary Banach spaces and T : X Y be a bounded linear operator. Let us recall that: (2) T is called (weakly) compact, if T(B X ) is a relatively (weakly) compact set in Y (see [1]). (3) T is (weakly) limited, if T(B X ) is (DP) limited set in Y (see [4, 14]). (4) T is completely continuous (DP), if it carries weakly null sequences in X to norm null sequences in Y (see [1, 18]). (5) T is limited completely continuous (abbreviated to Lcc), if T carries limited and weakly null sequences in X to norm null ones [18]. (6) A continuous operator T : E X from a Banach lattice to a Banach space is order weakly compact whenever T[0,x] is a relatively weakly compact subset of X for each x E + (see [1, p. 318]). Here, by introducing the concept of DPcc operators between Banach spaces, we obtain some characterizations of it and then the relation between the DPrcP of X and DP completely continuity of operators on X is treated. We obtain some sufficient conditions for the DPrcP of a closed subspace of some evaluation operator spaces, relative to DP complete continuity. Finally, We survey the domination of positive DPcc operators on Banach lattices. The notations and terminologies are normal. We use the symbols X, Y, Z for arbitrary Banach spaces and E, F for Banach lattices, respectively. We denoted the closed unit ball of X by B X, the dual of X by X, and T refers to the adjoint of the operator T. Also we use x,x for the duality between x X and x X. We refer the reader for undefined terminologies to the classical references [1, 6, 15, 17]. 1 Dunford-Pettis Completely Continuous Operators Let X and Y be arbitrary Banach spaces and T : X Y be a bounded linear operator. Since the DP weakly convergent sequences play an important role in the study of the DPrcP of Banach spaces, we define the following class of operators.

«, : Relatively Compact Dunford-Pettis Sets 3 Definition 1.1 T is called Dunford-Pettis completely continuous (DPcc), if T carries weakly null sequences which are DP sets in X to norm null ones. If T is DPcc, we call it a DPcc operator for convenience. We denote the class of all DP continuous operators from X to Y by DPcc(X,Y). Also, we use the symbols CC(X,Y) and Lcc(X, Y), for the class of all completely continuous and limited completely continuous operators from X to Y, respectively. Remark 1.1 It is obvious that DP operators are DP completely continuous. In general, the converse is false. On the other hand, every Dunford completely continuous operator is limited completely continuous, i.e., DPcc(X,Y) Lcc(X,Y). It is clear that the class DPcc(X,Y) is a closed linear subspace of L(X,Y), consisting of all bounded linear operators from X to Y; which has the ideal property, that is, for each T DPcc(X,Y) and each two bounded linear operators R and S, which can be composed with T, one has that RTS is also a DPcc operator. This implies that DPcc(X,Y) is an operator ideal in the sense of Pietsch [17]. Now, we give some characterizations of DPcc operators in terms of weakly limited and compact operators. Theorem 1.1 For a continuous operator T : X Y between two Banach spaces, the following statements are equivalent. (1) T is a DPcc operator. (2) T carries DP sets of X to relatively compact subsets of Y. (3) For an arbitrary Banach space Z and every weakly limited operator S : Z X, the operator TS is compact. (4) For every weakly limited operator S : l 1 X, the operator TS is compact. Proof (1) (2) Suppose that T : X Y is DPcc and A X is a DP set. Since every DP set isweaklyprecompact [10, p. 477], everysequence (x n ) in AhasaweakCauchysubsequence, denoted again by (x n ). On the other hand, the difference A A is a DP set. Thus, the sequence (x n x m ) is DP and weakly null, and then the sequence T(x n ) is a Cauchy sequence and so is norm convergent in Banach space Y. Hence, T(A) is relatively compact. (2) (3) Since S is weakly limited, so S(B Z ) is a DP set by the definition of weakly limited operator. Then by (2), TS(B Z ) is relatively compact. (3) (4) It is obvious. (4) (1) Let (x n ) be a weakly null DP sequence. Define the operator S : l 1 X by First, we claim that the set S(α 1,α 2, ) = { α n x n : n=1 n=1 α n x n. n=1 } α n 1

4 is a DP set. Suppose (f m ) to be a weakly null sequence in X. Then ( ) } sup{ f m α n x n : α n 1 n=1 n=1 = sup{ αn f m (x n ) : } α n 1 sup{ αn f m (x n ) : } α n 1 sup{ f m (x) : x (x n )} 0 as m. Thus it is a DP set. This implies that S is a weakly limited operator. For the basic unit sequence (e n ) of l 1, we have S(e n ) = x n. So TS(e n ) = T(x n ). w Clearly, TS(e n ) = Tx n 0 holds. Furthermore, by our hypothesis, TS is a compact operator. Hence (Tx n ) is a compact set. This implies that every subsequence of (Tx n ) has a subsequence converging in norm to zero. Therefore, we have Tx n 0. It is obvious that we can build a similar verdict about Lcc operator by observing the proof of Theorem 1.1. Now, we give the conclusion without proof. Theorem 1.2 For the continuous operator T : X Y between Banach spaces, the following are equivalent. (1) T is a Lcc operator; (2) T carries every limited set to relatively compact subset of Y; (3) For an arbitrary Banach space Z and every limited operator S : Z X, TS is compact; (4) For every limited operator S : l 1 X, TS is compact. Statement The equivalence between (1) and (2) is [18, Theorem 2.1]. In [9], the author stated that a dual Banach space with weak Radon-Nikodym property has the DPrcP. Any Banach space with the DPrcP has the so called compact range property [16]. In the following theorem, we give a characterization of the Banach space admitting the DPrcP, with respect to the DPcc operators. Theorem 1.3 For a Banach space X, the following are equivalent. (1) X has the DPrcP; (2) For each Banach space Y, DPcc(X,Y) = L(X,Y); (3) For each Banach space Y, DPcc(Y,X) = L(Y,X). Proof (1) (2) Assume that T L(X,Y) and that (x n ) is DP and weakly null. By our hypothesis, (x n ) is norm null. So Tx n 0; that is DPcc(X,Y) = L(X,Y). (2) (1) If Y = X, then (2) implies that the identity operator on X is DPcc. Otherwise, X has the DPrcP. The proof of the equivalence of (1) and (3) is similar. Remark 1.2 In fact, since the limited set is a DP set, so the DPrcP implies the GP property. Hence, if X has the DPrcP, by [18, Theorem 1.2], Lcc(X,Y) = L(X,Y) = DPcc(X,Y).

«, : Relatively Compact Dunford-Pettis Sets 5 Corollary 1.1 Every weakly compact operator is DPcc. Proof Let T : X Y be a weakly compact operator between Banach spaces X and Y. Since T carries DP sets into relatively compact sets [1, p. 350], by Theorem 1.1, T is a DPcc operator. For the following conclusion of Corollary 1.1, we first give a lemma from [1]. Keep in mind that totally bounded sets and relatively compact sets are equivalent in Banach spaces. Lemma 1.1 A subset A X is DP if and only if for each weakly compact operator X c 0, the subset S(A) of c 0 is norm totally bounded. Corollary 1.2 For an operator T : X Y, the following are equivalent. (1) T is weakly limited. (2) For each Banachspace Z and each weaklycompact operators : Y Z, the composition operator ST is relatively compact. (3) For each weakly compact operator S : Y c 0, the operator ST is relatively compact. Proof (1) (2) If T is weakly limited and S : Y Z is DPcc, then T(B X ) is a DP set of Y. By Corollary 1.1, every weakly compact operator is DPcc. By Theorem 1.1, S(T(B X )) is relatively compact. So the operator ST is relatively compact. (2) (3) It is obvious. (3) (1) For every operator S : Y c 0, the operator ST is relatively compact. That is, S(T(B X )) is a relatively compact set and so by Lemma 1.1, T(B X ) is DP in Y. The following theorem proves that the validity of the statement (2) of Theorem 1.3 by l instead of all Banach space Y, is a sufficient condition for the DPrcP of X. First, we give a lemma for the following proof. Lemma 1.2 A Banach space X has the DPrcP if and only if all weakly null DP sequences in X are norm null. Proof If X has the DPrcP, then the identity operator I : X X is a DPcc operator by Theorem 1.3. So for a weakly null DP sequence (x n ), x n = Ix n 0. Conversely, for an arbitrary Banach space Y, it is easy to know that every operator T L(X,Y) is a DPcc operator. Hence, by Theorem 1.3, X has the DPrcP. Theorem 1.4 A Banach space X has the DPrcP if and only if DPcc(X,l ) = L(X,l ). Proof If X does not have the DPrcP, then by Lemma 1.2, there is a weakly null DP sequence (x n ) in X such that x n = 1 for all n N. Choose a normalized sequence (x n) in X such that x,x n = 1 for all n, and define the operator T : X l by Tx = ( x,x n ), x X. But T is not DPcc, since the sequence (x n ) is DP and weakly null and Tx n 1 for all n. In Remark 1.1, we know that every completely continuous operator is DPcc, but the converse, in general, is false. In the following, we give a characterization of this converse assertion, with respect to the DP property of Banach spaces. First of all, we give a lemma for the convenience

6 of the following proof. The lemma is an easy conclusion of [3, Proposition 2.2] and [14, Theorem 2.4]. Lemma 1.3 A sequence (x n ) in X is DP if and only if x n,x n 0 for each weakly null sequence (x n) in X. Theorem 1.5 A Banach space X has the DP property if and only if for each Banach space Y, CC(X,Y) = DPcc(X,Y). Proof If X has the DP property and T : X Y is DPcc, then every weakly null sequence (x n ) inx isdp.byourhypothesisont, the sequence(tx n ) isnormnull. Hence, T iscompletely continuous. Conversely, if every DPcc operator on X is completely continuous, then X has the DP property. In fact, if (x n ) is a weakly null sequence but not a DP set in X, then by passing to a subsequence, there exists a weakly null sequence (x n ) in X such that x n,x n > ε, thanks to Lemma 1.3. Now, define the operator T : X c 0, i.e., Tx = ( x,x n ). By [1, Theorem 5.26], T is weakly compact, and by Corollary 1.1, T is DPcc. While it is not completely continuous, (x n ) is weakly null and for all n. This is a contradiction. Tx n x n,x n > ε Remark 1.3 In [1, Theorem 5.85], every AL and AM space has the DP property. So any two completely continuous and DPcc operator from AL or AM space to arbitrary Banach space coincide. In addition, if X has the DP property, then every DPcc operator from X into Y is a weak DP operator. In [12, Proposition 3.1], H Michane et al. proved that the Lcc operator from Banach lattice E to Banach space X is M-weakly compact when E has an order continuous norm and E has the DP* property. In fact, we have a similar conclusion about DPcc operators. Corollary 1.3 Let E be a Banach lattice and X be a Banach space. If E is a KB space and E has the DP property, then each DPcc operator T : M X is M-weakly compact. Proof Let T : M X be a DPcc operator and let (x n ) be a bounded disjoint sequence in E. By [1, Theorem 4.59], E has an order continuous norm. It follows from [7, Corollary 2.9] that x n w 0. On the other hand, since E has the DP property, then by Theorem 1.5, CC(E,X) = DPcc(E,X). By our hypothesis on T, we have Tx n 0, and thus T is M- weakly compact. We conclude this section by proving that the operator ideal DPCC between Banach spaces, by meaning of[5], is injective but not surjective. Recall that an operatorideal T is injective, if for each Banach space X,Y,Z and each isometric embedding J : Y Z, the operator T L(X,Y) belongs to T when JT T. Also T is surjective, if for each Banach spaces X,Y,Z and each surjection Q : Z X, the operator T L(X,Y) belongs to T when TQ T. The following theorem is similar to [18, Theorem 2.9] and its proof is also analogous.

«, : Relatively Compact Dunford-Pettis Sets 7 Theorem 1.6 The operator ideal DPCC is injective but not surjective. Proof Suppose that T L(X,Y) and J : Y Z is an isometric embedding, such that JT is DPcc. If (x n ) is DP and weakly null in X, then JTx n 0, as n. By our hypothesis on J, Tx n 0 and so T is DPcc. Hence, DPCC is injective. Now, for the proof of non-surjectivity of DPCC, suppose that X is a Banach space without the DPrcP. Then the identity operator i : X X is not DPcc. On the other hand, if one define Q X : l 1 (B X ) X via Q X (φ) = x B X φ(x)x, φ l 1 (B X ), then by [13], Q X is a surjective operator. Thus the Schur property and the DPrcP property of l 1 (B X ) imply that the operator Q X = iq X is DPcc, while the identity operator i is not. 2 Some Operator Spaces With the DPrcP In this section, we give some operator spaces which have the DPrcP. ForeachtwoBanachspacesX andy, bymeaningof[5]or[13], leti(x,y) bethe component of operators ideal I of all operators from X to Y that belongs to I. If M is a closed subspace of I(X,Y), for arbitrary elements x X and y Y, the evaluation operators φ x : M Y and ψ y : M X on M are defined by φ x (T) = Tx, ψ y (T) = T y, T M. Also, the point evaluation sets related to x X and y Y are the images of the closed unit ball B M of M, under the evaluation operators φ x and ψ y and are denoted by M(x) and M (y ) respectively. Theorem 2.1 For each two Banach spaces X and Y, if the closed subspace M of arbitrary operator ideal I(X,Y) has the DPrcP, then all evaluation operators φ x and ψ y are DPcc. Proof Since all φ x : M Y and ψ y : M X are bounded linear operators, it is an easy consequence of Theorem 1.3. By a similar method, we have the following necessary condition for the DPrcP of the dual of closed subspace M I(X,Y). Theorem 2.2 Suppose that X and Y have the DP property such that the dual M of a closed subspace M I(X,Y) has the DPrcP. Then all of the evaluation sets M(x) and M (y ) are DP sets. Proof Since M has the DPrcP, by Theorem 1.3, the adjoint operators φ x : Y M and ψ y : X M are DPcc. So by Theorem 1.5, these operators are completely continuous. For the proof M(x) Y is a DP set in Y, suppose that y n is a weakly null sequence in Y. Then φ x y n 0 as n for all x X. Since φ xy n = sup{ φ xy n,t : T B M } = sup{ y n,tx : T B M },

8 the sequence (yn ) converges uniformly on M(x). This shows that M(x) is a DP set in Y for all x X. A similar proof shows that M (y ) is a DP set in X. The following theorem is similar to [18, Theorem 3.8]. Theorem 2.3 Let X and Y be two Banach spaces such that Y has the Schur property. If M is a closed subspace of L(X,Y) such that each evaluation operators ψ y is DPcc on M, then M has the DPrcP. Proof If M does not have the DPrcP, then by Lemma 1.2, there is a weakly null DP sequence T n in M that is not norm null and by passing to a subsequence, we may assume that T n > ε for all integer n and some ε > 0. Choose a sequence (x n ) in B X such that T n x n > ε, for all n. In addition, for each y Y, the evaluation operator ψ y : M X is DPcc, so T n y = ψ y T n 0 and then T n x n,y T n y y 0. This means that the sequence (T n x n ) is weakly null and so norm null, thanks to the Schur property of Y. This contradiction shows that M has the DPrcP. By a similar method, we give a sufficient condition for the DPrcP of closed subspaces of L w (X,Y), consisting of all bounded weak -weak continuous operators from X to Y, and note that for each operator T L w (X,Y), the adjoint operator T maps Y into X. Theorem 2.4 Let X and Y be two Banach spaces such that X has the Schur property. If M is a closed subspace of L w (X,Y) such that each evaluation operators ψ y is DPcc on M, then M has the DPrcP. Proof Assume that one can choosea weaklynull DP sequence(t n ) in M such that T n > ε for some ε > 0 and all integer n. Then by our hypothesis on evaluation operators, for each x X, T n x = φ x (T n ) 0, as n. Since T n > ε, there exists a sequence (y n) in B Y such that T ny n > ε for all n. However, the Schur property of X implies that the weakly null sequence (Tn y n ) is norm null, which is a contradiction. In [11, Theorem 2], if X has the GP property and Y has the Schur property, then L(X,Y) has the GP property. Now, we have the similar conclusion about the DPrcP. Theorem 2.5 Let X have the DPrcP and Y have the Schur property. Then L(X,Y) has the DPrcP. Proof In order to prove that L(X,Y) has the DPrcP, it is enough to show that any weakly null DP sequence A n L(X,Y) is norm null by Lemma 1.2. We assume by contradiction that there is a DP sequence (A n ) such that A n w 0 and An = 1 for all n N. Let us consider y Y and the operator A A (y ) from L(X,Y) into X ; it is clear that A n(y ) w 0 and (A n (y )) is DP and so A n (y ) 0. So we have A n x n,y A n (y ) 0. From which it follows that A n (x n ) w 0. While Y has the Schur property, A n (x n ) 0, which is a contradiction. This completes the proof.

«, : Relatively Compact Dunford-Pettis Sets 9 The following corollary is similar to [11, Corollary 3]. Corollary 2.1 Let X have the Schur property and Y have the DPrcP. Then L(X,Y) has the DPrcP. Proof The mapping T T maps L(X,Y) onto a closed subspace of L(Y,X ), which has the DPrcP by virtue of Theorem 2.5. 3 Domination by Positive DPcc Operators on Banach Lattices In this section, we investigate the dominated property of DPcc operators on Banach lattices. Kalton-Saab [1,p.353] constructed a dominated theorem about DP operators (completely continuous operators). By the converse for the Kalton-Saab theorem proved by Wickstead [19], we know that there is a positive operator from L 1 [0,1] into c, which is dominated by a DP operator, is not DP. Since L 1 [0,1] is an AL-space, by [1, Theorem 5.85], it has the DP property. Thus, by Remark 1.3, there exists a positive operator from L 1 [0,1] into c dominated by a DPcc operator, which is not DPcc. Let us recall that a Banach lattice E has the weak DP property if every weakly compact operator T defined on E (take values in a Banach space X) is almost DP, that is, the sequence ( T(x n ) ) converges to zero in X for every weakly null sequence (x n ) consisting of pairwise disjoint elements in E (see [1, 3]), i.e., every weakly null sequence (x n ) consisting of pairwise disjoint elements in E is a DP set. Generally, we have the following domination result for the positive DPcc operators. Theorem 3.1 Let E, F be two Banach lattices and let S, T be two positive operators from E to F such that 0 S T and T is a DPcc operator. If E and F satisfy the following conditions: (1) E has the weak DP property; (2) F has order continuous norm, then S itself is a DPcc operator. Proof Suppose that F has an order continuous norm and that T : E F is a DPcc operator satisfying 0 S T. Let (x n ) be a weakly null and DP set, and let ε > 0. Put x = n=1 2 n x n, and let E x be the ideal generated by x in E. Also, let W denote the solid hull of the weakly relatively compact subset {x 1,x 2, } of E. Clearly, W E x holds. Next, note that if (y n ) is disjoint sequence of W, then (by [1, Theorem 4.34, p. 209]) we have y n w 0 in E. Since E has the weak DP property, (y n ) is a DP set. Since T is a DPcc operator, so Ty n 0. Thus, by [1, Theorem 4.36, p. 210], there exists some 0 u E x such that holds for all n. T( x n u) + < ε Next, consider the operator S,T : E x F. Then, by [1, Theorem 4.87, p. 268], there exist

10 operators M 1,M 2,,M k on E x and positive operators L 1,L 2,,L k on F satisfying k S L i TM i u ε and on E x. 0 k L i TM i T Since each M i : E x E x is continuous for the norm induced by E, it is easy to see that M i (x n ) w 0 holds in E and (M i (x n )) is DP. Thus, using the fact that T is a DPcc operator, we see that lim k L itm i (x n ) = 0. Pick some m such that k L itm i (x n ) < ε holds for all n m. Now note that for n m we have ( k k Sx n S L i TM i )x n + L i TM i (x n ) k S L i TM i ( x n u) + k + S L i TM i u +ε and so Sx n 0 holds. 2 T( x n u) + +ε+ε < 4ε The next result is analogousto [1, Theorem 5.93, p. 347] which is due to Kalton-Saab, while it is valid for DPcc operators. Theorem 3.2 Let E and F be two Banach lattices with E having the weak DP property. Consider the scheme of operators E S1 F S2 X, where S 1 is a positive operatorand dominated by a DPcc oprator. If S 2 is order weakly compact, then S 2 S 1 is a DPcc operator. Proof By [1, Theorem 5.58, p. 319], the operator S 2 admits a factorization through a Banach lattice G with order continuous norm F Q G S X such that Q is a lattice homomorphism and S 2 = QS. Obviously, the positive operator QS 1 : E G is dominated by a DPcc operator. Thus, by Theorem 3.1, the operator QS 1 is also DPcc, and consequently S 2 S 1 = S(QS 1 ) is likewise a DPcc operator. For the next conclusion of Theorem 3.2, we first give a lemma. Lemma 3.1 Every DPcc operator T : E X from a Banach lattice to a Banach space is order weakly compact. Proof Let (x n ) be an order bounded disjoint sequence of E. Then x n w 0 holds in E. On the other hand, (x n ) is a DP set by [3, Theorem 2.5]. So lim Tx n 0. The conclusion now follows immediately from [1, Theorem 5.57, p. 318].

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