Properties (𝑉) and (𝑤𝑉) in Projective Tensor Products

Abstract

We give sufficient conditions for a subset of to be relatively weakly compact. A Banach space has property (resp., ) if every -subset of is relatively weakly compact (resp., weakly precompact). We prove that the projective tensor product has property (resp., ), when has property (resp., ), has property , and .

1. Introduction

Throughout this paper, , and  will denote real Banach spaces. An operator will be a continuous and linear function. The set of all continuous linear transformations from  to  will be denoted by and the compact operators will be denoted by .

In this paper, we study weak precompactness and relative weak compactness in spaces of compact operators. Our results are organized as follows. First, we give sufficient conditions for subsets of to be weakly precompact and relatively weakly compact. These results are used to study whether the projective tensor product has properties and , when and have the respective property. Finally, we prove that, in some cases, if has property , then .

2. Definitions and Notations

Our notation and terminology are standard. We denote the canonical unit vector basis of by and the canonical unit vector basis of by . The unit ball of will be denoted by unless otherwise specified, and will denote the continuous linear dual of . The set of all continuous linear transformations from to will be denoted by and the compact and weakly compact operators will be denoted by , respectively . The operator is completely continuous (or Dunford-Pettis) if maps weakly Cauchy sequences to norm convergent sequences.

A subset of is said to be weakly precompact provided that every bounded sequence from has a weakly Cauchy subsequence. The Banach space has the Dunford-Pettis property () if every weakly compact operator is completely continuous.

A series in is said to be weakly unconditionally convergent (wuc) if, for every , the series is convergent. An operator is unconditionally converging if it maps weakly unconditionally convergent series to unconditionally convergent ones.

A bounded subset of is called a -subset of provided that for each wuc series in .

Pełczyński introduced property in his fundamental paper [1]. The Banach space has property if every -subset of is relatively weakly compact. The following results were also established in [1]: reflexive Banach spaces and spaces have property ; the Banach space has property if and only if every unconditionally converging operator from to any Banach space is weakly compact; every quotient space of a Banach space with property has property ; if has property , then is weakly sequentially complete.

A Banach space is said to have property weak if any -subset of is weakly precompact [2]. If , then has property , by Rosenthal’s theorem ([3], Ch. XI).

A topological space is called dispersed (or scattered) if every nonempty closed subset of has an isolated point. A compact Hausdorff space is dispersed if and only if [4].

A Banach space has the approximation property if, for each norm compact subset of and , there is a finite rank operator such that for all . If in addition can be found with , then is said to have the metric approximation property. spaces, , , , ( any measure), , and their duals have the metric approximation property [5, 6].

A separable Banach space has an unconditional compact expansion of the identity (u.c.e.i) if there is a sequence of compact operators from to such that converges unconditionally to for all [7]. In this case, is called an (u.c.e.i.) of . A sequence of closed subspaces of a Banach space is called an unconditional Schauder decomposition of if every has a unique representation of the form , with , for every , and the series converges unconditionally [8].

The space has (Rademacher) cotype for some if there is a constant such that, for every and every in , where are the Rademacher functions. A Hilbert space has cotype 2. The dual of , , has cotype 2 [9].

The Banach-Mazur distance between two isomorphic Banach spaces and is defined by , where the infimum is taken over all isomorphisms from onto . A Banach space is called an -space (resp. -space) [10] if there is a so that every finite dimensional subspace of is contained in another subspace with (resp., ) for some integer . Complemented subspaces of spaces (resp., spaces) are -spaces (resp., -spaces) (Proposition 1.26, [10]). The dual of an -space (resp., -space) is an -space (resp., -space) (Proposition 1.27, [10]). The -spaces, -spaces, and their duals have the (Corollary 1.30, [10]).

3. Weakly Precompact Subsets of Spaces of Compact Operators

We begin by giving sufficient conditions for a subset of to be weakly precompact and relatively weakly compact. We recall that the dual weak operator topology on is defined by the functional , , and [11].

If , , , and , let , , and .

Theorem 1. Let be a subset of such that (i) is weakly precompact compact for all ;(ii) is relatively weakly compact for all .Then is weakly precompact.

Proof. Suppose that is a subset of satisfying (i) and (ii). Let be a sequence in and let be the closed linear span of . Note that is a separable subspace of , since is compact for each . Let be a countable subset of that separates . For , is weakly precompact. By diagonalization, we can (and do) assume that is weakly Cauchy for all (by passing to a subsequence if necessary).
Let . By hypothesis, is relatively weakly compact. Let and be two weak sequential cluster points of . Note that , since . Assume that and . For , (since is weakly Cauchy). Hence , since is a separating set of . Let such that , for each . Then for all , is weakly Cauchy in and thus is weakly Cauchy in . Hence is Cauchy in the topology on . By Corollary 3 of [11], is weakly Cauchy in . Therefore, is weakly precompact.

Using a similar proof, we can obtain the following theorem.

Theorem 2. Let be a subset of such that (i) is weakly precompact compact for all ;(ii) is relatively weakly compact for all .Then is weakly precompact.

Proof. Suppose that is a subset of satisfying (i) and (ii). Let be a sequence in and let be the closed linear span of . Note that is a separable subspace of  , since is compact for each . Let be a countable subset of  that separates . For , is weakly precompact. By diagonalization, we can (and do) assume that is weakly Cauchy for all (by passing to a subsequence if necessary).
Let . By hypothesis, is relatively weakly compact. Let and be two weak sequential cluster points of . Since is compact, , and hence . Assume that and . For , Hence , since is a separating set of . Let such that , for each . Then is weakly Cauchy in for all , and hence is Cauchy in the topology on . By Corollary 3 of  [11], is weakly Cauchy in and thus is weakly Cauchy in . Therefore, is weakly precompact.

Corollary 3. (i) Suppose that and is reflexive. Then .
(ii) Suppose that and is reflexive. Then .

Proof. (i) Let be a sequence in , . For each , is weakly precompact, since . For each , is relatively weakly compact, since is reflexive. By Theorem 2, has a weakly Cauchy subsequence. By Rosenthal’s theorem, contains no copies of .
(ii) By (i), , hence contains no copies of .

Theorem 4. Suppose that and is a subset of such that(i) is relatively weakly compact for all ;(ii) is relatively weakly compact for all .Then is relatively weakly compact.

Proof. By Theorem 1, is weakly precompact. Let be a sequence in . Without loss of generality, assume that is weakly Cauchy. For each , the sequence has a weakly convergent subsequence and is weakly Cauchy and hence is weakly convergent. Similarly, for each , the sequence has a weakly convergent subsequence and is weakly Cauchy and thus is weakly convergent.
Let such that for all . Then for all . Since is weakly convergent, for all . Hence in the topology on . By Corollary 3 of  [11], , and is relatively weakly compact.

Corollary 5 (see [6], Theorem 4.19). Suppose that and are reflexive. If , then and are reflexive.

Proof. Let be the unit ball of . Since and are reflexive, and are relatively weakly compact for all and . By Theorem 4, is relatively weakly compact. Then is reflexive; thus is reflexive.

4. Properties and in Projective Tensor Products

In this section, we consider properties and in the projective tensor product . We begin by noting that there are examples of Banach spaces and such that has property . If , then [12]. If is the conjugate of , then is reflexive (by [6], Theorem 4.19) and thus has property [1]. Then the spaces and are as desired.

Observation 1. If is an infinite dimensional space with the Schur property, then does not have property . This can be seen since , and thus ([3], page 211). All bounded subsets of are -sets; thus there are -subsets of which fail to be weakly precompact.

Since property is inherited by quotients, it follows that if has property , then .

Lemma 6. If is weakly unconditionally convergent in and is weakly unconditionally convergent in , then is weakly unconditionally convergent in .

Proof. Let . Suppose is wuc in and is wuc in . Since is wuc in and is wuc in , , by Corollary 2 of  [13].

Observation 2. If is an operator such that is (weakly) compact, then is (weakly) compact. To see this, let be an operator such that is (weakly) compact. Let . Suppose and choose a net in which is -convergent to . Then . Now, , which is a relatively (weakly) compact set. Then (resp., ). Hence, , which is relatively (weakly) compact. Therefore, is relatively (weakly) compact; thus is (weakly) compact.

It follows that if , then and if , then .

Observation 3. If has property and has property , then every operator is weakly compact. To see this, let be an operator. Since has property , (by Observation 1) and [14]. Then is unconditionally converging [14]. Hence is weakly compact, since has property [1].

Theorem 7. (i) Suppose that has property , has property , and . Then has property .
(ii) Suppose that has property , has property , and . Then has property .

Proof. (i) Since has property and has property , (by Observation 3). Therefore, . Let be a -subset of and let be a sequence in .
Let . We show that is a -subset of . Suppose that is wuc in . If , , since is unconditionally convergent. Then is wuc in . Since is a -set, Therefore, is a -subset of and thus is weakly precompact.
Let . We show that is a -subset of . Let be wuc in . For ,
It is enough to show that is weakly null. Suppose that is wuc in . By Lemma 6, is wuc in . Since is a -set, Therefore, is a -subset of and thus is weakly precompact. We can assume that is weakly Cauchy. If , since is a -set and is wuc in . Hence is -null. Since is also weakly Cauchy, is weakly null. Then is a -subset of and thus is relatively weakly compact. By Theorem 1, is weakly precompact.
(ii) If , then (by Observation 2). By (i), , thus has property .

Theorem 8 (Theorem 2, [15]). Suppose that and have property and . Then has property .

Proof. Since and have property , (by Observation 3). Therefore, . Let be a -subset of and let be a sequence in . Since and have property , and are both weakly sequentially complete [1]; thus is weakly sequentially complete, by Theorem 3.10 of  [16]. By Theorem 7, we can (and do) assume that is weakly Cauchy. Then is weakly convergent, since is weakly sequentially complete.

Remark 9. Theorem 8 can be proved using Theorems 7 and 4. To see this, let be a -subset of and let be a sequence in . The proof of Theorem 7 shows that, for all , is a -subset of , and thus is relatively weakly compact. Similarly, for all , is a -subset of and thus is relatively weakly compact. By Theorem 4, has a weakly convergent subsequence.

Corollary 10 (Corollary 4, [15]). Suppose that and , where . Then has property .

Proof. Note that and have property [1]. If is the conjugate of , then . Every operator , , is compact ([17], page 100). Apply Theorem 8.

Corollary 11. (i) Suppose has property , has property , and has the Schur property. Then has property .
(ii) Suppose that has property , has the Schur property, and has property . Then has property .
(iii) (Corollary 5, [15]) Suppose and have property . If or has the Schur property, then has property .

Proof. (i) Since is a Schur space, . Apply Theorem 7.
(ii) Let be weakly compact. Then is weakly compact and thus is compact, since has the Schur property. Hence is compact (by Observation 2) and . Apply Theorem 7.
(iii) If or has the Schur property, then . Apply Theorem 8.

The space satisfies the hypothesis (i) when has property , and has the and does not contain . Note that has the Schur property by Theorem 3 of [18]. In particular, we can take , with dispersed ().

Let  be the original James space [19]. Since is separable and 1-codimensional in , all duals of are separable and fails to embed in any of them. Moreover, none of these spaces can be weakly sequentially complete. Thus and its duals have the and property , but none of these spaces have property (since their duals are not weakly sequentially complete).

If or its duals and is dispersed, then the space has property (by Corollary 11) and does not have property (since does not have property ).

The fact that properties and are inherited by quotients immediately implies the following result, which contains Corollary 3 of  [15].

Corollary 12. Suppose that , has property (resp., property ), and has property . Then the space of all nuclear operators from to has property (resp., property ).

Proof. It is known that is a quotient of ([6], page 41). By Theorem 7 (resp., Theorem 8), has property (resp., ). Hence has property (resp., property ).

Theorem 13. Suppose that . The following statements are equivalent:(i) and have property and or ,(ii) has property .

Proof. (i) ⇒ (ii) Apply Theorem 8.
(ii) ⇒ (i) Suppose that has property . Then and have property , since property is inherited by quotients. We will show that or . Suppose that and . Hence [20]. Also, the Rademacher functions span inside of ; thus . Similarly . Then ([21, 22]). Thus [14], a contradiction with Observation 1.

Observation 4. If and , then , , and ([21, 22]). More generally, if , , and , then ([21, 22]). Thus [14], and does not have property (by Observation 1).

Corollary 14. Suppose that and has property . The following statements are equivalent:(i) has property and or ,(ii) has property .

Proof. (i) ⇒ (ii) Apply Theorem 7.
(ii) ⇒ (i) If has property , then has property , since property is inherited by quotients. Apply Observation 4.

The next result contains Corollary 6 of [15].

Corollary 15. Suppose that and have the . The following statements are equivalent:(i) and have property (resp., property ) and or ,(ii) has property (resp., property ).

Proof. (i) ⇒ (ii) Suppose that and have the . Without loss of generality suppose . Then has the Schur property [18]. Suppose that and have property . Apply Corollary 11(iii). Now suppose that and have property . Since has property and is weakly sequentially complete, has property . Apply Corollary 11(ii). (ii) ⇒ (i) by Observation 4.

By Corollary 15, the space has property if and only if either or is dispersed. The spaces and have the and property and contain copies of ([23–26]). Let , be or . Then does not have property (by Observation 4).

Next we present some results about the necessity of the condition .

Theorem 16. Suppose has property . Then , provided any one of the following holds.(i)(Theorem 7, [15]) Either or has the metric approximation property.(ii)If is an operator which is not compact, then there is a sequence in such that, for each , the series converges unconditionally to .(iii)Either or has an (u.c.e.i.).(iv)(Theorem 8, [15]) is complemented in a Banach space which has an unconditional Schauder decomposition , and for all .(v)(Theorem 7, [15]) or has the compact approximation property and is a closed subspace of a Banach space possessing an (u.c.e.i.).(vi)(Theorem 7, [15]) Either or has an (u.c.e.i.).

Proof. Suppose has property . Then is weakly sequentially complete [1].
(i) If or has the metric approximation property and is weakly sequentially complete, then by Corollary 2.4 of [27].
(ii) Let be a noncompact operator. Let be a sequence as in the hypothesis. By the uniform boundedness principle, is bounded in . Then is wuc and not unconditionally convergent (since is noncompact). Hence [14] and we have a contradiction.
(iii) Suppose that has an (u.c.e.i.) . Then is compact for each and converges unconditionally to , for each . Let be a noncompact operator. Hence converges unconditionally to for each and . Then (by (ii)), and we have a contradiction.
Similarly, if has an (u.c.e.i.) and , then .
(iv) Let be a weakly compact and noncompact operator, , , and let be the projection of onto . Define by , , . Note that is compact since . Then is compact for each . Since, for each , converges unconditionally to , converges unconditionally to for each . By (ii), , and we have a contradiction.
(v) If , then (by Corollary 2.5 of [7]), a contradiction.
(vi) If , then (by Theorem 6 of [11]), a contradiction.
Thus, . Since and have property , (by Observation 3). Hence .

The proof of Theorem 16 and Observation 4 show that if one of the hypotheses (ii)–(vi) holds and has property , then . Hence , where and and are conjugates, does not have property , since the natural inclusion map is weakly compact and not compact. The space does not have property , if is not dispersed and (by Observation 4, since and ).

By Theorem 6 of [11], ; thus . Since , for ([6]), (by Proposition 2.4 of [6]). Thus , for . Hence , with , does not have property (by Observation 1).

Theorem 17. Assume one of the following holds.(i) is an -space and is a subspace of an -space.(ii), is a compact Hausdorff space, and is a space with cotype 2.(iii) or has the Schur property.(iv) has the and .(v) and have the .If has property , then .

Proof. Suppose has property . Then and have property .
Suppose (i) or (ii) holds. It is known that any operator is 2-absolutely summing ([28]); hence it factorizes through a Hilbert space. If , then (by Remark 3 of [29]), a contradiction with Observation 1.
(iii) Suppose that has the Schur property. Since has property and is weakly sequentially complete, has property . By Observation 3, . Since has the Schur property, (see the proof of Corollary 11). Hence .
Similarly, if has the Schur property, then . By Observation 2, .
(iv) By Observation 4, . Since also has the , has the Schur property ([18]). Apply (iii).
(v) If , then (iv) implies . If , then has the Schur property ([18]); thus (by (iii)).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.