The (𝐷) Property in Banach Spaces

Abstract

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_∞. Studying this property in connection with other geometric properties, we show that every Banach space whose dual has (V^∗) property of Pełczyński (and hence every Banach space with (V) property) has (D) property. We show that the space 𝐿¹(𝑣) of real functions, which are integrable with respect to a measure 𝑣 with values in a Banach space 𝑋, has (D) property. We give some other results concerning Banach spaces with (D) property.

1. 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 and Ülger in [4], and, then, further results on the weak Phillips property was given in a paper by Ülger [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, 𝐿¹(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 𝐿¹-spaces have (𝐷) property. This is a structural difference between 𝐶(𝐾) spaces (which enjoy (𝑉) property) and 𝐿¹-spaces (which fail to have (𝑉) property).

2. 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 𝑙₁ and abstract 𝐿-spaces enjoy the (𝑉^∗) property, whereas, the space 𝑐₀ 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 function ‖𝑣‖(𝐴)=sup{|𝑥^∗𝑣|(𝐴)∶𝑥^∗∈𝐵_𝑋∗}, where |𝑥^∗𝑣| is the variation of the scalar measure 𝑥^∗𝑣. A Rybakov control measure for 𝑣 is a measure 𝜆=|𝑥^∗𝑣| such that 𝜆(𝐴)=0 if and only if ‖𝑣‖(𝐴)=0 (see [8]).

Following Lewis [7], we will say that a measurable function 𝑓∶Ω→𝐼𝑅 is integrable with respect to 𝑣 if(1)𝑓 is 𝑥^∗𝑣 integrable for every 𝑥^∗∈𝑋^∗,(2)for each 𝐴∈Σ there exists an element of 𝑋 denoted by ∫_𝐴𝑓𝑑𝑣, such that 𝑥^∗_𝐴𝑓𝑑𝑣=_𝐴𝑓𝑑𝑥^∗𝑣forevery𝑥^∗∈𝑋^∗. (2.1)

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 𝑓_𝑣=sup_𝐴𝑓𝑑𝑥^∗𝑣∶𝑥^∗∈𝐵_𝑋∗, (2.2) becomes a Banach space. We will denote it by 𝐿¹(𝑣). It is a Banach lattice for the ‖𝑣‖-almost everywhere order. Simple functions are dense in 𝐿¹(𝑣) and the identity is a continuous injection of the space of ‖𝑣‖-essentially bounded functions into 𝐿¹(𝑣).

3. 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_𝑛𝑘) equivalent to the 𝑙₁ unit vector basis so that the closed linear space [(𝑥_𝑛𝑘)] is complemented in 𝐸.

Now, we give the following lemma, which we need for the same operations done in Ülger's paper [5].

Lemma 3.2. Let 𝐸 be a Banach space and 𝐹 a dual Banach space not containing an isomorphic copy of 𝑙_∞. Then, 𝑇∶𝐸^∗→𝐹 is unconditionally converging.

Proof. For a contradiction, suppose that we have an operator 𝑇∶𝐸^∗→𝐹 that is unconditionally converging for a Banach space 𝐸 and a dual Banach space 𝐹 not containing an isomorphic copy of 𝑙_∞. Then, there exists a subspace 𝑀 of 𝐸^∗ such that 𝑀 is isomorphic to 𝑐₀ and the restriction 𝑇∣_𝑀 of T is an isomorphism on 𝑀 by [10]. Since 𝑀≅𝑐₀, we have 𝑀^∗∗≅𝑙_∞. Considering the natural injection 𝑖∶𝑀→𝐸^∗ and the natural projection 𝑝∶𝐸^∗∗∗→𝐸^∗, take the composition 𝑇𝑜𝑝𝑜𝑖^∗∗∶𝑀^∗∗→𝑌. Let us denote it as 𝑆=𝑇𝑜𝑝𝑜𝑖^∗∗. Since 𝑀^∗∗≅𝑙_∞ is an injective space [11] and every operator from an injective space to a space not containing an isomorphic copy of 𝑙_∞ is weakly compact from Corollary  1.4 of Rosenthal [12], 𝑆 is weakly compact. Then the restriction 𝑆∣_𝑀=(𝑇𝑜𝑝𝑜𝑖^∗∗)∣_𝑀 is weakly compact. Since 𝑆∣_𝑀=(𝑇𝑜𝑝𝑜𝑖^∗∗)∣_𝑀=𝑇∣_𝑀 holds, 𝑇∣_𝑀 is weakly compact, which means (𝑇∣_𝑀)^∗∗(𝑀^∗∗)⊂𝑐₀ by Goldstein theorem. Since the restriction 𝑇∣_𝑀 is an isomorphism, a contradiction occurs that means 𝑙_∞⊂𝑐₀. Then, 𝑇∶𝐸^∗→𝐹 is unconditionally converging.

Theorem 3.3. Let 𝐸 be a Banach space whose dual has the (𝑉^∗) property. Then, 𝐸 has the (𝐷) property.

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. If a Banach space 𝑋 has the (𝑉) property, then it has the (𝐷) property.

Corollary 3.4 shows that the space 𝑐₀ 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^∗≅𝑙₁ and the space 𝑙₁ 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 𝑐₀, it fails to have the (𝑉) property.
As a commonly known example, the space 𝑙₁ does not have (𝐷) property because the injection 𝐼∶𝑐₀→𝑙_∞ is not weakly compact. Recall that the dual space of 𝑐₀, that is, 𝑙₁, 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 𝑙₁.
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.

Corollary 3.7. Let 𝐸 be a Banach space having the (𝐷) property. Then, 𝐸 has (𝑊) property.

Proof. Let 𝐸 be a Banach space having the (𝐷) property. Then, 𝐸 cannot contain a complemented subspace that is isomorphic to 𝑙₁, 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 well-known 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 𝑙₁.

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_𝑛𝑘))] isomorphic to 𝑙₁. It is easy to see that [(𝑦_𝑛𝑘)] must be a complemented subspace of 𝐹 isomorphic to 𝑙₁.
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 Theorem 3.10, which indicates another structural difference between 𝐶(𝐾) spaces (which enjoy (𝑉) property) and 𝐿¹-spaces (which fail to have (𝑉) property).

Theorem 3.10. The space 𝐿¹(𝑣) has (𝐷) property.

Proof. Let 𝐹 be a Banach space whose dual does not contain an isomorphic copy of 𝑙_∞, and, a bounded operator 𝑇∶𝐹→𝐿¹(𝑣)^∗ be given. Let 𝜆 be a Rybakov control measure for 𝑣. Then, 𝐿¹(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 𝐿¹(𝜆), and the dual space 𝐿¹(𝑣)^∗ can be identified with the space of functions 𝑔 in 𝐿¹(𝜆) such that 𝑓𝑔∈𝐿¹(𝜆), for all 𝑓 in 𝐿¹(𝑣), where the action of 𝑔 over 𝐿¹(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 𝐿¹(𝜆). Then, by Dunford's theorem in [8], we have a weakly convergence subsequence (𝑇(𝑓_𝑛𝑘)), which means that the operator 𝑇 is weakly compact. Hence, 𝐿¹(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 [3–5]. 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].

Proposition 3.11. The James space 𝐽 fails to have the (𝐷) property.

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 𝑙₁, by Rosenthal's 𝑙₁ theorem, (𝑦_𝑛) has a weakly Cauchy subsequence (𝑦_𝑛𝑗). Then, considering the adjoint operator S^∗∶𝐽^∗∗→𝐹^∗, for each 𝑥^′′∈𝐽^∗∗, we see that 𝑆^∗(𝑥^′′)∈𝐹^∗. By the equality ⟨𝑦_𝑛𝑗,𝑆^∗(𝑥^′′)⟩=⟨𝑆(𝑦_𝑛𝑗),𝑥^′′⟩ for 𝑥^′′∈𝐽^∗∗, the subsequence (𝑆(𝑦_𝑛𝑗)) becomes a weakly Cauchy subsequence in 𝐽^∗. For yielding a contradiction, using the same technique in [19], suppose that (𝑆(𝑦_𝑛𝑗)) does not converge weakly in 𝐽^∗; then by using [19, Lemma  6] we have a subsequence (𝑆(𝑥_𝑛𝑗)) of (𝑆(𝑦_𝑛𝑗)) and a constant 𝑀<∞ such that 𝑐_𝑗𝑣^′_𝑗≤𝑀𝑐_𝑗𝑆𝑥_𝑛𝑗⋯ (3.1) 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_𝑛𝑗)=𝑥_𝑛𝑗 for all 𝑗. Therefore, by [19] 𝑐_𝑗𝑣^′_𝑗≤𝑀𝑐_𝑗𝑆𝑥_𝑛𝑗≤𝑀‖𝑆‖𝑐_𝑗𝑥_𝑛𝑗≤𝑀‖𝑆‖‖𝑅‖𝑐_𝑗𝑥_𝑛𝑗 (3.2) for all finitely supported real sequence (𝑐_𝑗). But this means that the operator 𝑇 from [19, Proposition  3] is bounded, yielding a contradiction. Hence, (𝑆(𝑦_𝑛𝑗)) converges weakly in 𝐽^∗, which means that the identity operator 𝑆∶𝐽^∗→𝐽^∗ is weakly compact. However, this case is impossible. Therefore, the James space 𝐽 fails to have (𝐷) property.
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 𝑙₁.

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 𝑙₁. 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 𝑙₁ [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*-to-norm (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.