Abstract and Applied Analysis
Volume 2012 (2012), Article ID 212957, 8 pages
http://dx.doi.org/10.1155/2012/212957
Research Article

A New Class of Banach Spaces and Its Relation with Some Geometric Properties of Banach Spaces

Department of Mathematics, Yazd University, Yazd 89195-741, Iran

Received 29 December 2011; Accepted 14 February 2012

Copyright © 2012 M. Salimi and S. M. Moshtaghioun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By introducing the concept of L-limited sets and then L-limited Banach spaces, we obtain some characterizations of it with respect to some well-known geometric properties of Banach spaces, such as Grothendieck property, Gelfand-Phillips property, and reciprocal Dunford-Pettis property. Some complementability of operators on such Banach spaces are also investigated.

1. Introduction and Preliminaries

A subset of a Banach space is called limited (resp., Dunford-Pettis (DP)), if every weak* null (resp., weak null) sequence in converges uniformly on , that is, Also if and every weak null sequence in converges uniformly on , we say that is an L-set.

We know that every relatively compact subset of is limited and clearly every limited set is DP and every DP subset of a dual Banach space is an L-set, but the converse of these assertions, in general, are false. If every limited subset of a Banach space is relatively compact, then has the Gelfand-Phillips property (GP). For example, the classical Banach spaces and have the GP property and every reflexive space, every Schur space (i.e., weak and norm convergence of sequences in coincide), and dual of spaces containing no copy of have the same property.

Recall that a Banach space is said to have the DP property if every weakly compact operator is completely continuous (i.e., maps weakly null sequences into norm null sequences) and is said to have the reciprocal Dunford-Pettis property (RDP) if every completely continuous operator on is weakly compact.

So the Banach space has the DP property if and only if every relatively weakly compact subset of is DP and it has the RDP property if and only if every L-set in is relatively weakly compact.

A stronger version of DP property was introduced by Borwein et al. in [1]. In fact, a Banach space has the DP* property if every relatively weakly compact subset of is limited. But if is a Grothendieck space (i.e., weak and weak* convergence of sequences in coincide), then these properties are the same on . The reader can find some useful and additional properties of limited and DP sets and Banach spaces with the GP, DP, or RDP property in [26].

We recall from [7] that a bounded linear operator is limited completely continuous (lcc) if it carries limited and weakly null sequences in to norm null ones in . We denote the class of all limited completely continuous operators from to by . It is clear that every completely continuous operator is lcc and we showed in [7] that every weakly compact operator is limited completely continuous.

Here, by introducing a new class of subsets of Banach spaces that are called L-limited sets, we obtain some characterizations of Banach spaces that every L-limited set is relatively weakly compact and then we investigate the relation between these spaces with the GP, DP, RDP and Grothendieck properties.

The notations and terminologies are standard. We use the symbols , , and for arbitrary Banach spaces. We denoted the closed unit ball of by , absolutely closed convex hull of a subset of by , the dual of by , and refers to the adjoint of the operator . Also we use for the duality between and . We denote the class of all bounded linear, weakly compact, and completely continuous operators from to by , , and respectively. We refer the reader for undefined terminologies to the classical references [8, 9].

2. L-Limited Sets

Definition 2.1. A subset of dual space is called an L-limited set, if every weak null and limited sequence in converges uniformly on .

It is clear that every L-set in is L-limited and every subset of an L-limited set is the same. Also, it is evident that every L-limited set is weak* bounded and so is bounded. The following theorem gives additional properties of these sets.

Theorem 2.2. (a) Absolutely closed convex hull of an L-limited set is L-limited.
(b) Relatively weakly compact subsets of dual Banach spaces are L-limited.
(c) Every weak* null sequence in dual Banach space is an L-limited set.

Proof. Let be an L-limited set, and the sequence in is weak null and limited. Since the proof of (a) is clear. For the proof of (b) suppose is relatively weakly compact but it is not an L-limited set. Then there exists a weakly null and limited sequence in , a sequence in and an such that for all integer . Since is relatively weakly compact, there exists a subsequence of that converges weakly to an element . Since we have a contradiction.
Finally, for (c), suppose is a weak* null sequence in . Define the operator by . Since has the GP property by [7], is lcc. So for each weakly null and limited sequence in , we have as . Hence is an L-limited set.

Note that the converse of assertion (b) in general is false. In fact, the following theorem show that the closed unit ball of is an L-limited set, while the standard unit vectors in , as a weakly null sequence, shows that the closed unit ball of is neither an L-set nor a relatively weakly compact. The following Theorem 2.4, give a necessary and sufficient condition for Banach spaces that L-sets and L-limited sets in its dual coincide.

Theorem 2.3. A Banach space has the GP property if and only if every bounded subset of is an L-limited set.

Proof. Since the Banach space has the GP property if and only if every limited and weakly null sequence in is norm null [10], the proof is clear.

Theorem 2.4. A Banach space has the property if and only if each L -limited set in is an L-set.

Proof. Suppose has the DP* property. Since every weakly null sequence in is limited so every L-limited set in is L-set.
Conversely, it is enough to show that, for each Banach space , [7, Theorem 2.8]. If is lcc, it is clear that is an L-limited set. So by hypothesis, it is an L-set and we know that the operator is completely continuous if and only if is an L-set.

The following two corollaries extend Theorem 3.3 and Corollary 3.4 of [1].

Corollary 2.5. For a Banach space , the following are equivalent.(a) has the DP* property,(b)If has the Gelfand-Phillips property, then each operator is completely continuous.

Proof. (a) (b). Suppose that has the Gelfand-Phillips property. By [7, Theorem 2.2], every operator is lcc, thus is an L-limited set and by Theorem 2.3, it is an L-set. Hence is completely continuous.
(b) (a). If does not have the DP* property, there exists a weakly null sequence in that is not limited. So there is a weak * null sequence in such that , for all integer and some positive [10]. Now the bounded operator defined by is not completely continuous, since is weakly null and for all . This is a contradiction.

Corollary 2.6. A Gelfand-Phillips space has the DP* property if and only if it has the Schur property.

Proof. It is clear that the Banach space has the Schur property if and only if every bounded subset of is an L-set. Now, if is a GP space with the DP* property, then by Theorem 2.3, unit ball is L-limited and so it is an L-set. The converse of the assertion is also clear.

Definition 2.7. A Banach space has the L-limited property, if every L-limited set in is relatively weakly compact.

Theorem 2.8. For a Banach space , the following are equivalent:(a) has the L-limited property,(b)for each Banach space , ,(c).

Proof. (a) (b). Suppose that has the L-limited property and is lcc. Thus is an L-limited set in . So by hypothesis, it is relatively weakly compact and is a weakly compact operator.
(b) (c). It is clear.
(c) (a). If does not have the L-limited property, there exists an L-limited subset of that is not relatively weakly compact. So there is a sequence with no weakly convergent subsequence. Now we show that the operator defined by for all is limited completely continuous but it is not weakly compact. As is L-limited set, for every weakly null and limited sequence in we have thus is a limited completely continuous operator. It is easy to see that , for all . Thus is not a weakly compact operator and neither is . This finishes the proof.

The following corollary shows that the Banach spaces and do not have the L-limited property.

Corollary 2.9. A Gelfand-Phillips space has the L-limited property if and only if it is reflexive.

Proof. If a Banach space has the GP property, then by [7], the identity operator on is lcc and so is weakly compact, thanks to the L-limited property of . Hence is reflexive.

Theorem 2.10. If a Banach space has the L-limited property, then it has the RDP and Grothendieck properties.

Proof. At the first, we show that has the RDP property. For arbitrary Banach space , let be a completely continuous operator. Thus it is limited completely continuous and so by Theorem 2.8, is weakly compact. Hence has the RDP property.
By [11], we know that a Banach space is Grothendieck if and only if . Since has the GP property, by [7], and by hypothesis on , . So is Grothendieck.

We do not know the converse of Theorem 2.10, in general, is true or false. In the following, we show that in Banach lattices that are Grothendieck and have the DP property, the converse of this theorem is correct.

Theorem 2.11. If a Banach lattice has both properties of Grothendieck and DP, then it has the L-limited property.

Proof. Suppose that is limited completely continuous. We know, that in Grothendieck Banach spaces, DP and DP* properties are equivalent. Thus by [7], is completely continuous. On the other hand, is not a Grothendieck space and Grothendieck property is carried by complemented subspaces. Hence the Grothendieck space does not have any complemented copy of . Since is a Banach lattice, by [12], it has the RDP property and so the completely continuous operator is weakly compact. Thus has the L-limited property, thanks to Theorem 2.8.

As a corollary, since is a Banach lattice that has Grothendieck and DP properties, it has the L-limited property. This shows that the L-limited property on Banach spaces is not hereditary, since does not have this property. In the following, we show that the L-limited property is carried by every complemented subspace.

Theorem 2.12. If a Banach space has the L-limited property, then every complemented subspace of has the L-limited property.

Proof. Consider a complemented subspace of and a projection map . Suppose is a limited completely continuous operator, so is also lcc. Since has the L-limited property, by Theorem 2.8, is weakly compact. Hence is weakly compact.

As another corollary, for infinite compact Hausdorff space , we have the following corollary for the Banach space of all continuous functions on with supremum norm.

Corollary 2.13. has the L-limited property if and only if it contains no complemented copy of .

Proof. We know that is a Banach lattice with the DP property. On the other hand, is a Grothendieck space if and only if it contains no complemented copy of [13]. So the direct implication is an application of Theorem 2.12 and the opposite implication is also an easy conclusion of Theorem 2.11.

3. Complementation in Lcc Operators

In [11], Bahreini investigated the complementability of and in . She showed that if is not a reflexive Banach space, then is not complemented in and if is not a Schur space, is not complemented in . In the following, we investigate the complementability of and in . We need the following lemma of [14].

Lemma 3.1. Let be a separable Banach space, and is a bounded linear operator with for all . Then there is an infinite subset of such that for each , , where is the set of all with for each .

Theorem 3.2. If does not have the L-limited property, then is not complemented in .

Proof. Consider a subset of that is L-limited but it is not relatively weakly compact. So there is a sequence in that has no weakly convergent subsequence. Hence defined by is an lcc operator but it is not weakly compact. Choose a bounded sequence in such that has no weakly convergent subsequence. Let , the closed linear span of the sequence in . It follows that is a separable subspace of such that is not a weakly compact operator. If , we have is bounded and has no weakly convergent subsequence.
Now define by , where and . Then
We claim that for each , .
Fix and a weakly null and limited sequence in . Since is an L-limited set, . So we have as . This finishes the proof of the claim and so is a well-defined operator into .
Let be the restriction map and define Now suppose that is complemented in and is a projection. Define by . Note that as is a rank one operator, we have . Hence for all . By Lemma 3.1, there is an infinite set so that for all . Thus is a weakly compact operator. On the other hand, if is the standard unit vectors of , for each and each , we have Therefore for all . Thus is not a weakly compact operator and neither is . This contradiction ends the proof.

Corollary 3.3. Let be a Banach space. Then the following are equivalent:(a) has the L-limited property,(b),(c) is complemented in .

We conclude this paper with another complementation theorem. Recall from [11] that a closed operator ideal has the property (*) whenever is a Banach space and is not in , then there is an infinite subset such that is not in for all infinite subsets , where is the operator defined by , for all .

Theorem 3.4. If a Banach space does not have the DP* property, then is not complemented in .

Proof. By hypothesis, there is a weakly null sequence in that is not limited. So there exists a weak* null sequence in such that Now define the operator by . By Theorem 2.2, is an L-limited set, but is not completely continuous. So for , is not completely continuous. Since has the property (*) [11, Theorem 4.12], one can choose so that for each infinite subset of , . Define by , where and . As shown in the proof of the preceding theorem, is well defined.
Let be the restriction map and define Now suppose is complemented in and is a projection. Define by . Since for all , one can use Lemma 3.1 to select an infinite subset of such that for all . Thus belongs to for each . But , so we have a contradiction.

Corollary 3.5. Let be a Banach space. Then the following are equivalent:(a) has the DP* property,(b),(c) is complemented in .

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