The (D) Property in Banach Spaces

and Applied Analysis 3 1 f is x∗v integrable for every x∗ ∈ X∗, 2 for each A ∈ Σ there exists an element of X denoted by ∫A fdv, such that


Introduction
As it is well known, the properties or forms that remain invariant under a group of transformations are called "geometry." The geometry of Banach spaces covers all of the properties that do not change under isomorphisms. We can collect these properties in two groups, namely, isometric and isomorphic ones. Isometric properties such as strict or smooth convexity are directly connected to the norms, whereas, isomorphic ones like Schur property or V property of Pełczyński depend on the topologies that the norms define, rather than the norms themselves. In the last half-century, defining new geometric properties of Banach spaces and studying them have gained great interest 1 . The reason for these developments is that examining the structure of Banach spaces with the help of these properties is easier than investigating them one by one. In the literature, there has been a plenty of geometric properties defined in Banach spaces so far. As very illuminative tools, Pełczyński's fundamental paper 2 introducing the so-called u , V , and V * properties and the second paper 3 that defined the V and V * properties by the coincidence of V or V * sets with the weakly relatively compact sets can be given as examples. Many important Banach spaces properties are or can be defined in the same way, that is, by the coincidence of two classes of bounded sets. As an example, Phillips and weak Phillips properties were introduced by Freedman andÜlger in 4 , and, then, further results on the weak Phillips property was given in a paper byÜlger 5 .
In this paper, we introduce a new kind of geometric properties in Banach spaces as follows. Let E and F be two Banach spaces and E * the dual space of E. A Banach space E is said to have D property if every bounded linear operator T : F → E * is weakly compact for every Banach space F whose dual does not contain an isomorphic copy of l ∞ . We show that every Banach space whose dual has V * property has D property. We show that V property implies D property; however, the converse implication does not hold. We also see that D property implies W property. We investigate some properties of such Banach spaces to some extent; for instance, we show that the James space J fails to have the D property.
Given a vector measure ν with values in a Banach space X, L 1 v denotes the space of classes of real functions that are integrable with respect to v in the sense of Bartle et al. 6 and Lewis 7 . We show that L 1 -spaces have D property. This is a structural difference between C K spaces which enjoy V property and L 1 -spaces which fail to have V property .

Notations and Preliminaries
We will try to follow the standard notations in the Banach space theory. In order to prevent any doubt, we will fix some terminology. If E is a Banach space, B E will be its closed unit ball and E * its topological dual. The word operator will always mean linear bounded operator. A series x n in E is said to be weakly unconditionally Cauchy w.u.c. in short if |x x n | < ∞ for every x ∈ E * . An operator T : E → F is said to be unconditionally converging if T sends w.u.c. series x n in E into unconditionally converging series Tx n in F. An operator T : E → F is said to be weakly compact if T B E is relatively weakly compact. It is well known that every weakly compact operator is unconditionally converging 3 .
A Banach space E is said to have V property if every unconditionally converging operator T : E → F is weakly compact for every Banach space F.
A Banach space E is said to have V * property if, for every Banach space F, every operator T : F → E is weakly compact whenever its adjoint T * : E * → F * is unconditionally converging.
The above definitions of V and V * spaces were firstly introduced by Pełczyński, who showed that the space l 1 and abstract L-spaces enjoy the V * property, whereas, the space c 0 and C K spaces enjoy the V property 3 . Now, we introduce our following definition of D property in Banach spaces.
A Banach space E is said to have D property if every bounded linear operator T : F → E * is weakly compact for every Banach space F whose dual does not contain an isomorphic copy of l ∞ .
Let Ω, Σ be a measurable space, X a Banach space with unit ball B X and dual space X * , and v : Σ → X a countably additive vector measure. The semivariation of v is the set Lewis 7 , we will say that a measurable function f : Identifying two functions if the set where they differ has null semivariation, we obtain a linear space of classes of functions that, when endowed with the norm becomes a Banach space. We will denote it by L 1 v . It is a Banach lattice for the v -almost everywhere order. Simple functions are dense in L 1 v and the identity is a continuous injection of the space of v -essentially bounded functions into L 1 v .

Main Results on the D Property
We start with Lemma 3.1., which is adapted from a result given by Godefroy and Saab in 9 .
Lemma 3.1. Let E be a Banach space with the V * property and x n a bounded but not relatively compact sequence in E. Then, there is a subsequence x n k equivalent to the l 1 unit vector basis so that the closed linear space x n k is complemented in E.
Now, we give the following lemma, which we need for the same operations done in Ulger's paper 5 .  Proof. Let E be a Banach space whose dual has the V * property and F a Banach space whose dual does not contain an isomorphic copy of l ∞ . Let an operator T : F → E * be given.
Consider the adjoint operator T * : E * * → F * . Then, by Lemma 3.2, the operator T * : E * * → F * is unconditionally converging. By the definition of V * property, the operator T : F → E * is weakly compact. If any Banach space X enjoys the V property, then dual space X * has the V * property; therefore, we immediately have the following result. Corollary 3.4 shows that the space c 0 and C K spaces have the D property. But, we give the following example to show that the converse of Corollary 3.4 does not hold.
Example 3.5. Let Y be the space constructed by Bourgain and Delbaen in 13 . Since Y * ∼ l 1 and the space l 1 has the Schur property, the space Y has D property. However, since the space Y is not reflexive and does not contain a copy of c 0 , it fails to have the V property.
As a commonly known example, the space l 1 does not have D property because the injection I : c 0 → l ∞ is not weakly compact. Recall that the dual space of c 0 , that is, l 1 , does not contain a copy of l ∞ . By the definition of D property it is clear that any Banach space with D property does not contain a complemented copy of l 1 .
Any Banach space with the V * property is weakly sequentially complete w.s.c. and every closed subspace of a w s.c. space is also w.s.c.; so such space does not contain an isomorphic copy of l ∞ . Recall that the space l ∞ is not w.s.c.. If a Banach space has the V property, then its dual has the V * property 3 . Hence, Corollary 3.4 extends a previous result of E. Saab and P. Saab 14 , which we give as a corollary below. Corollary 3.6. Let E be a Banach space with the V property. Then, every operator T : E → E * is weakly compact [14].
E. Saab and P. Saab, in 14 , introduced a property called the W property. A Banach space X has the W property if every operator T : E → E * is weakly compact. Proof. Let E be a Banach space having the D property. Then, E cannot contain a complemented subspace that is isomorphic to l 1 , and E * cannot contain a subspace isomorphic to l ∞ . It follows that every operator T : E → E * is weakly compact, that is, E has the W property.
Taking into consideration Theorem 3.3 and Corollary 3.7, we have the following wellknown result. Corollary 3.8. Let the dual space E * has the V * property. Then, E has W property. Theorem 3.9. Let E be a Banach space whose dual has the V * property and F any Banach space. If any bounded linear operator T : F → E * is not weakly compact, then T fixes an isomorphic copy of l 1 .
Proof. Let E * have the V * property. Suppose that T : F → E * is not weakly compact. Then, there is a bounded sequence y n in F so that T y n is not relatively weakly compact. Hence, from Lemma 3.1, a subsequence generates a complemented subspace T y n k isomorphic to l 1 . It is easy to see that y n k must be a complemented subspace of F isomorphic to l 1 .
The example we gave after Corollary 3.4 shows that D property does not imply V property is not, of course, the unique counterexample. For example, we want to give Abstract and Applied Analysis 5 Theorem 3.10, which indicates another structural difference between C K spaces which enjoy V property and L 1 -spaces which fail to have V property . Proof. Let F be a Banach space whose dual does not contain an isomorphic copy of l ∞ , and, a bounded operator T : F → L 1 v * be given. Let λ be a Rybakov control measure for v. Then, L 1 v is an order continuous Banach function space with weak unit over the finite measure space Ω, Σ , λ see 15 . Thus, it can be regarded as a lattice ideal in L 1 λ , and the dual space L 1 v * can be identified with the space of functions g in L 1 λ such that fg ∈ L 1 λ , for all f in L 1 v , where the action of g over L 1 v is given by integration with respect to λ 16 . Suppose that f n is a bounded sequence in F; then T f n can be considered as a bounded set in L 1 λ . Then, by Dunford's theorem in 8 , we have a weakly convergence subsequence T f n k , which means that the operator T is weakly compact. Hence, L 1 v has the D property.
It is known that dual spaces of the Banach spaces having the geometric properties such as V , weak Phillips, or Grothendieck properties are weakly sequentially complete; see [3][4][5] . At this point the following question arises: does the dual of any Banach space with D property have to be weakly sequentially complete, or not? We could not give an answer to this question yet. Regarding D property, we now just give the following theorem that reveals a result opposite to our expectation about the James space J, which is not weakly sequentially complete as shown in 17 . Proof. By recalling the construction of the James space as in 18 , we will give a proof using contradiction method as follows: suppose the James space J has the D property. Take F as J * and any bounded sequence y n in F. Since F does not contain any isomorphic copy of l 1 , by Rosenthal's l 1 theorem, y n has a weakly Cauchy subsequence y n j . Then, considering the adjoint operator S * : J * * → F * , for each x ∈ J * * , we see that S * x ∈ F * . By the equality y n j , S * x S y n j , x for x ∈ J * * , the subsequence S y n j becomes a weakly Cauchy subsequence in J * . For yielding a contradiction, using the same technique in 19 , suppose that S y n j does not converge weakly in J * ; then by using 19, Lemma 6 we have a subsequence S x n j of S y n j and a constant M < ∞ such that for every finitely supported real sequence c j . According to 19, Proposition 5 we may assume that there is a bounded linear operator R : F → F such that R u n j x n j for all j. Therefore, by 19 6 Abstract and Applied Analysis The James space J fails to have D property by Proposition 3.7, and J is not weakly sequentially complete because it is not reflexive and does not contain any isomorphic copy of l 1 . Theorem 3.12. Any nonreflexive space E with E * * not containing l ∞ does not have the D property.
Proof. Let E be a nonreflexive space E with E * * not containing l ∞ . For the contradiction, suppose that E enjoys the D property and G is a subspace of E * that does not contain a complemented copy of l 1 . Since the condition that F * does not contain a copy of l ∞ is equivalent to the condition that F does not contain a complemented copy of l 1 17 , the embedding i : G → E * is weakly compact. Hence, G is reflexive. That is, the identity operator I : E * → E * is weakly compact, which is a contradiction.
Before raising some questions, we want to recall some geometric properties in the literature we think can be related with the property D . A Banach space E is said to have the weak Phillips property if the natural projection p : E * * * → E * is sequentially weak * -tonorm weak * -to-weak continuous 4, 5 . A Banach space E is said to have the Grothendieck property if, for every separable Banach space F, every operator T : E → F is weakly compact 8 .

Open Questions
We do not know whether D property implies or is implied by either the weak Phillips property or the Grothendieck property. We do not know whether weak sequential completeness of any dual Banach space E * implies that the space E itself has D property. Also, we do not know whether D property of any Banach space implies V * property of its dual space.