# Properties(V) and (w

Authors: Ioana Ghenciu
Journal: Journal of Operators
 (2014)

---

## Abstract

We give sufficient conditions for a subset ofK
(
X
,
Y
*
)
=
L
(
X
,
Y
*
) to be relatively weakly compact. A Banach space X has property (
V
) (resp., (
w
V
)) if every V-subset of X
* is relatively weakly compact (resp., weakly precompact). We prove that the projective tensor product X
⊗
π
Y has property (
V
) (resp., (
w
V
)), when X has property (
V
) (resp., (
w
V
)), Y has property (
V
), and W
(
X
,
Y
*
)
=
K
(
X
,
Y
*
).


---

## Body

## 1. Introduction

Throughout this paper,X
,
Y
,
E, and F  will denote real Banach spaces. An operator T
:
X
→
Y will be a continuous and linear function. The set of all continuous linear transformations from X  to Y  will be denoted by L
(
X
,
Y
) and the compact operators will be denoted by K
(
X
,
Y
).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 ofK
(
X
,
Y
*
) to be weakly precompact and relatively weakly compact. These results are used to study whether the projective tensor product X
⊗
π
Y has properties (
w
V
) and (
V
), when X and Y have the respective property. Finally, we prove that, in some cases, if X
⊗
π
Y has property (
V
), then L
(
X
,
Y
*
)
=
K
(
X
,
Y
*
).


## 2. Definitions and Notations

Our notation and terminology are standard. We denote the canonical unit vector basis ofc
0 by (
e
n
) and the canonical unit vector basis of l
1 by (
e
n
*
). The unit ball of X will be denoted by B
X unless otherwise specified, and X
* will denote the continuous linear dual of X. The set of all continuous linear transformations from X to Y will be denoted by L
(
X
,
Y
) and the compact and weakly compact operators will be denoted by K
(
X
,
Y
), respectively W
(
X
,
Y
). The operator T is completely continuous (or Dunford-Pettis) if T maps weakly Cauchy sequences to norm convergent sequences.A subsetS of X is said to be weakly precompact provided that every bounded sequence from S has a weakly Cauchy subsequence. The Banach space X has the Dunford-Pettis property (DPP) if every weakly compact operator T
:
X
→
Y is completely continuous.A series∑
‍
x
n in X is said to be weakly unconditionally convergent (wuc) if, for every x
*
∈
X
*, the series ∑
‍
|
x
*
(
x
n
)
| is convergent. An operator T
:
X
→
Y is unconditionally converging if it maps weakly unconditionally convergent series to unconditionally convergent ones.A bounded subsetA of X
* is called a V-subset of X
* provided that
(1)
lim
⁡
n
(
sup
⁡
{
|
x
*
(
x
n
)
|
:
x
*
∈
A
}
)
=
0
for each wuc series ∑
‍
x
n in X.Pełczyński introduced property(
V
) in his fundamental paper [1]. The Banach space X has property (
V
) if every V-subset of X
* is relatively weakly compact. The following results were also established in [1]: reflexive Banach spaces and C
(
K
) spaces have property (
V
); the Banach space X has property (
V
) if and only if every unconditionally converging operator T from X to any Banach space Y is weakly compact; every quotient space of a Banach space with property (
V
) has property (
V
); if X has property (
V
), then X
* is weakly sequentially complete.A Banach spaceX is said to have property weak (
V
)
(
w
V
) if any V-subset of X
* is weakly precompact [2]. If l
1
↪
X
*, then X has property (
w
V
), by Rosenthal’s theorem ([3], Ch. XI).A topological spaceS is called dispersed (or scattered) if every nonempty closed subset of S has an isolated point. A compact Hausdorff space K is dispersed if and only if l
1
↪
C
(
K
) [4].A Banach spaceX has the approximation property if, for each norm compact subset M of X and ϵ
>
0, there is a finite rank operator T
:
X
→
X such that ∥
T
(
x
)
-
x
∥
<
ϵ for all x
∈
M. If in addition T can be found with ∥
T
∥
≤
1, then X is said to have the metric approximation property. C
(
K
) spaces, c
0, l
p, 1
≤
p
<
∞, L
p
(
μ
) (μ any measure), 1
≤
p
<
∞, and their duals have the metric approximation property [5, 6].A separable Banach spaceX has an unconditional compact expansion of the identity (u.c.e.i) if there is a sequence (
A
n
) of compact operators from X to X such that ∑
‍
A
n
(
x
) converges unconditionally to x for all x
∈
X [7]. In this case, (
A
n
) is called an (u.c.e.i.) of X. A sequence (
X
n
) of closed subspaces of a Banach space X is called an unconditional Schauder decomposition of X if every x
∈
X has a unique representation of the form x
=
∑
‍
x
n, with x
n
∈
X
n, for every n, and the series converges unconditionally [8].The spaceX has (Rademacher) cotype q for some 2
≤
q
≤
∞ if there is a constant C such that, for every n and every x
1
,
x
2
,
…
,
x
n in X,
(2)
(
∑
i
=
1
n
‍
∥
x
i
∥
q
)
1
/
q
≤
C
(
∫
0
1
‍
∥
r
i
(
t
)
x
i
∥
q
d
t
)
1
/
q
,
where (
r
n
) are the Rademacher functions. A Hilbert space has cotype 2. The dual of C
(
K
), M
(
K
), has cotype 2 [9].The Banach-Mazur distanced
(
E
,
F
) between two isomorphic Banach spaces E and F is defined by inf
⁡
(
∥
T
∥
∥
T
-
1
∥
), where the infimum is taken over all isomorphisms T from E onto F. A Banach space E is called an L
∞-space (resp. L
1-space) [10] if there is a λ
≥
1 so that every finite dimensional subspace of E is contained in another subspace N with d
(
N
,
l
∞
n
)
≤
λ (resp., d
(
N
,
l
1
n
)
≤
λ) for some integer n. Complemented subspaces of C
(
K
) spaces (resp., L
1
(
μ
) spaces) are L
∞-spaces (resp., L
1-spaces) (Proposition 1.26, [10]). The dual of an L
1-space (resp., L
∞-space) is an L
∞-space (resp., L
1-space) (Proposition 1.27, [10]). The L
∞-spaces, L
1-spaces, and their duals have the DPP (Corollary 1.30, [10]).


## 3. Weakly Precompact Subsets of Spaces of Compact Operators

We begin by giving sufficient conditions for a subset ofK
(
X
,
Y
*
) to be weakly precompact and relatively weakly compact. We recall that the dual weak operator topology (
w
′
) on L
(
X
,
Y
) is defined by the functional T
→
x
*
*
T
*
(
y
*
), x
*
*
∈
X
*
*, and y
*
∈
Y
* [11].IfH
⊆
L
(
X
,
Y
*
), x
∈
X, x
*
*
∈
X
*
*, and y
*
*
∈
Y
*
*, let H
(
x
)
=
{
T
(
x
)
:
T
∈
H
}, H
*
(
y
*
*
)
=
{
T
*
(
y
*
*
)
:
T
∈
H
}, and H
*
*
(
x
*
*
)
=
{
T
*
*
(
x
*
*
)
:
T
∈
H
}.Theorem 1.
LetH be a subset of K
(
X
,
Y
*
) such that (i)
H
(
x
) is weakly precompact compact for all x
∈
X;
(ii)
H
*
(
y
*
*
) is relatively weakly compact for all y
*
*
∈
Y
*
*.Then H is weakly precompact.Proof.
Suppose thatH is a subset of K
(
X
,
Y
*
) satisfying (i) and (ii). Let (
T
n
) be a sequence in H and let S be the closed linear span of {
T
n
*
(
y
*
*
)
:
y
*
*
∈
Y
*
*
,
n
∈
N
}. Note that S is a separable subspace of X
*, since T
n is compact for each n
∈
N. Let X
0 be a countable subset of X that separates S. For x
∈
X
0, (
T
n
(
x
)
) is weakly precompact. By diagonalization, we can (and do) assume that (
T
n
(
x
)
) is weakly Cauchy for all x
∈
X
0 (by passing to a subsequence if necessary).
Lety
*
*
∈
Y
*
*. By hypothesis, (
T
n
*
(
y
*
*
)
) is relatively weakly compact. Let z
1 and z
2 be two weak sequential cluster points of (
T
n
*
(
y
*
*
)
). Note that z
1
,
z
2
∈
S, since T
n
*
(
Y
*
*
)
⊂
S. Assume that T
p
(
n
)
*
(
y
*
*
)
→
w
z
1 and T
k
(
n
)
*
(
y
*
*
)
→
w
z
2. For x
∈
X
0,
(3)
z
1
(
x
)
=
lim
⁡
n
〈
T
p
(
n
)
*
(
y
*
*
)
,
x
〉
=
lim
⁡
n
〈
y
*
*
,
T
p
(
n
)
(
x
)
〉
=
lim
⁡
n
〈
y
*
*
,
T
n
(
x
)
〉
=
lim
⁡
n
〈
y
*
*
,
T
k
(
n
)
(
x
)
〉
=
lim
⁡
n
〈
T
k
(
n
)
*
(
y
*
*
)
,
x
〉
=
z
2
(
x
)
(since (
T
n
(
x
)
) is weakly Cauchy). Hence z
1
=
z
2, since X
0 is a separating set of S. Let L
:
Y
*
*
→
S such that T
n
*
(
y
*
*
)
→
w
L
(
y
*
*
), for each y
*
*
∈
Y
*
*. Then for all y
*
*
∈
Y
*
*, (
T
n
*
(
y
*
*
)
) is weakly Cauchy in S and thus is weakly Cauchy in X
*. Hence (
T
n
) is Cauchy in the (
w
′
) topology on K
(
X
,
Y
*
). By Corollary 3 of [11], (
T
n
) is weakly Cauchy in K
(
X
,
Y
*
). Therefore, H is weakly precompact.Using a similar proof, we can obtain the following theorem.Theorem 2.
LetH be a subset of K
(
X
,
Y
*
) such that (i)
H
*
(
y
) is weakly precompact compact for all y
∈
Y;
(ii)
H
*
*
(
x
*
*
) is relatively weakly compact for all x
*
*
∈
X
*
*.Then H is weakly precompact.Proof.
Suppose thatH is a subset of K
(
X
,
Y
*
) satisfying (i) and (ii). Let (
T
n
) be a sequence in H and let S be the closed linear span of {
T
n
(
x
)
:
x
∈
X
,
n
∈
N
}. Note that S is a separable subspace of  Y
*, since T
n is compact for each n
∈
N. Let Y
0 be a countable subset of  Y that separates S. For y
∈
Y
0, (
T
n
*
(
y
)
) is weakly precompact. By diagonalization, we can (and do) assume that (
T
n
*
(
y
)
) is weakly Cauchy for all y
∈
Y
0 (by passing to a subsequence if necessary).
Letx
*
*
∈
X
*
*. By hypothesis, (
T
n
*
*
(
x
*
*
)
) is relatively weakly compact. Let z
1 and z
2 be two weak sequential cluster points of (
T
n
*
*
(
x
*
*
)
). Since T
n is compact, T
n
*
*
(
X
*
*
)
⊂
S, and hence z
1
,
z
2
∈
S. Assume that T
p
(
n
)
*
*
(
x
*
*
)
→
w
z
1 and T
k
(
n
)
*
*
(
x
*
*
)
→
w
z
2. For y
∈
Y
0,
(4)
z
1
(
y
)
=
lim
⁡
n
〈
T
p
(
n
)
*
*
(
x
*
*
)
,
y
〉
=
lim
⁡
n
〈
x
*
*
,
T
p
(
n
)
*
(
y
)
〉
=
lim
⁡
n
〈
x
*
*
,
T
n
*
(
y
)
〉
=
lim
⁡
n
〈
x
*
*
,
T
k
(
n
)
*
(
y
)
〉
=
lim
⁡
n
〈
T
k
(
n
)
*
*
(
x
*
*
)
,
y
〉
=
z
2
(
y
)
.
Hence z
1
=
z
2, since Y
0 is a separating set of S. Let L
:
X
*
*
→
S such that T
n
*
*
(
x
*
*
)
→
w
L
(
x
*
*
), for each x
*
*
∈
X
*
*. Then (
T
n
*
*
(
x
*
*
)
) is weakly Cauchy in S for all x
*
*
∈
X
*
*, and hence (
T
n
) is Cauchy in the (
w
′
) topology on K
(
X
,
S
). By Corollary 3 of  [11], (
T
n
) is weakly Cauchy in K
(
X
,
S
) and thus is weakly Cauchy in K
(
X
,
Y
*
). Therefore, H is weakly precompact.Corollary 3.
(i) Suppose thatl
1
↪
X
* and Y is reflexive. Then l
1
↪
K
(
X
,
Y
*
).
(ii) Suppose thatl
1
↪
Y
* and X is reflexive. Then l
1
↪
K
(
X
,
Y
*
).Proof.
(i) Let(
T
n
) be a sequence in K
(
X
,
Y
*
), ∥
T
n
∥
≤
1. For each y
∈
Y, (
T
n
*
(
y
)
) is weakly precompact, since l
1
↪
X
*. For each x
*
*
∈
X
*
*, (
T
n
*
*
(
x
*
*
)
) is relatively weakly compact, since Y is reflexive. By Theorem 2, (
T
n
) has a weakly Cauchy subsequence. By Rosenthal’s l
1 theorem, K
(
X
,
Y
*
) contains no copies of l
1.
(ii) By (i),K
(
Y
,
X
*
), hence K
(
X
,
Y
*
) contains no copies of l
1.Theorem 4.
Suppose thatL
(
X
,
Y
*
)
=
K
(
X
,
Y
*
) and H is a subset of L
(
X
,
Y
*
) such that(i)
H
(
x
) is relatively weakly compact for all x
∈
X;
(ii)
H
*
(
y
*
*
) is relatively weakly compact for all y
*
*
∈
Y
*
*.Then H is relatively weakly compact.Proof.
By Theorem1, H is weakly precompact. Let (
T
n
) be a sequence in H. Without loss of generality, assume that (
T
n
) is weakly Cauchy. For each x
∈
X, the sequence (
T
n
(
x
)
) has a weakly convergent subsequence and is weakly Cauchy and hence is weakly convergent. Similarly, for each y
*
*
∈
Y
*
*, the sequence (
T
n
*
(
y
*
*
)
) has a weakly convergent subsequence and is weakly Cauchy and thus is weakly convergent.
LetT
∈
L
(
X
,
Y
*
)
=
K
(
X
,
Y
*
) such that (
T
n
(
x
)
)
→
w
T
(
x
) for all x
∈
X. Then (
T
n
*
(
y
*
*
)
)
→
w
*
T
*
(
y
*
*
) for all y
*
*
∈
Y
*
*. Since (
T
n
*
(
y
*
*
)
) is weakly convergent, (
T
n
*
(
y
*
*
)
)
→
w
T
*
(
y
*
*
) for all y
*
*
∈
Y
*
*. Hence (
T
n
)
→
T in the (
w
′
) topology on K
(
X
,
Y
*
). By Corollary 3 of  [11], (
T
n
)
→
w
T, and H is relatively weakly compact.Corollary 5 (see [6], Theorem 4.19).
Suppose thatX and Y are reflexive. If L
(
X
,
Y
*
)
=
K
(
X
,
Y
*
), then L
(
X
,
Y
*
) and X
⊗
π
Y are reflexive.Proof.
LetH be the unit ball of L
(
X
,
Y
*
). Since X and Y are reflexive, H
(
x
) and H
*
(
y
*
*
) are relatively weakly compact for all x
∈
X and y
*
*
∈
Y
*
*. By Theorem 4, H is relatively weakly compact. Then (
X
⊗
π
Y
)
*
≃
L
(
X
,
Y
*
) is reflexive; thus X
⊗
π
Y is reflexive.


## 4. Properties(
w
V
) and (
V
) in Projective Tensor Products

In this section, we consider properties(
V
) and (
w
V
) in the projective tensor product X
⊗
π
Y. We begin by noting that there are examples of Banach spaces X and Y such that X
⊗
π
Y has property (
V
). If 1
<
q
′
<
p
<
∞, then L
(
l
p
,
l
q
′
)
=
K
(
l
p
,
l
q
′
) [12]. If q is the conjugate of q
′, then l
p
⊗
π
l
q is reflexive (by [6], Theorem 4.19) and thus has property (
V
) [1]. Then the spaces X
=
l
p and Y
=
l
q are as desired.Observation 1.
IfX is an infinite dimensional space with the Schur property, then X does not have property (
w
V
). This can be seen since l
1
↪
X, and thus l
1
↪
X
* ([3], page 211). All bounded subsets of X
* are V-sets; thus there are V-subsets of X
* which fail to be weakly precompact.Since property(
w
V
) is inherited by quotients, it follows that if X has property (
w
V
), then l
1
↪
c
X.Lemma 6.
If∑
‍
x
n is weakly unconditionally convergent in X and ∑
‍
y
n is weakly unconditionally convergent in Y, then ∑
‍
x
n
⊗
y
n is weakly unconditionally convergent in X
⊗
π
Y.Proof.
LetT
∈
(
X
⊗
π
Y
)
*
≃
L
(
X
,
Y
*
). Suppose ∑
‍
x
n is wuc in X and ∑
‍
y
n is wuc in Y. Since ∑
‍
T
(
x
n
) is wuc in Y
* and ∑
‍
y
n is wuc in Y, ∑
‍
|
〈
T
,
x
n
⊗
y
n
〉
|
=
∑
‍
|
〈
T
(
x
n
)
,
y
n
〉
|
<
∞, by Corollary 2 of  [13].Observation 2.
IfT
:
Y
→
X
* is an operator such that T
*
|
X is (weakly) compact, then T is (weakly) compact. To see this, let T
:
Y
→
X
* be an operator such that T
*
|
X is (weakly) compact. Let S
=
T
*
|
X. Suppose x
*
*
∈
B
X
*
* and choose a net (
x
α
) in B
X which is w
*-convergent to x
*
*. Then (
T
*
(
x
α
)
)
→
w
*
T
*
(
x
*
*
). Now, (
T
*
(
x
α
)
)
⊆
S
(
B
X
), which is a relatively (weakly) compact set. Then (
T
*
(
x
α
)
)
→
T
*
(
x
*
*
) (resp., (
T
*
(
x
α
)
)
→
w
T
*
(
x
*
*
)). Hence, T
*
(
B
X
*
*
)
⊆
S
(
B
X
)
¯, which is relatively (weakly) compact. Therefore, T
*
(
B
X
*
*
) is relatively (weakly) compact; thus T is (weakly) compact.It follows that ifL
(
X
,
Y
*
)
=
K
(
X
,
Y
*
), then L
(
Y
,
X
*
)
=
K
(
Y
,
X
*
) and if L
(
X
,
Y
*
)
=
W
(
X
,
Y
*
), then L
(
Y
,
X
*
)
=
W
(
Y
,
X
*
).Observation 3.
IfX has property (
V
) and Y has property (
w
V
), then every operator T
:
X
→
Y
* is weakly compact. To see this, let T
:
X
→
Y
* be an operator. Since Y has property (
w
V
), l
1
↪
c
Y (by Observation 1) and c
0
↪
Y
* [14]. Then T is unconditionally converging [14]. Hence T is weakly compact, since X has property (
V
) [1].Theorem 7.
(i) Suppose thatX has property (
V
), Y has property (
w
V
), and W
(
X
,
Y
*
)
=
K
(
X
,
Y
*
). Then X
⊗
π
Y has property (
w
V
).
(ii) Suppose thatX has property (
w
V
), Y has property (
V
), and W
(
X
,
Y
*
)
=
K
(
X
,
Y
*
). Then X
⊗
π
Y has property (
w
V
).Proof.
(i) SinceX has property (
V
) and Y has property (
w
V
), L
(
X
,
Y
*
)
=
W
(
X
,
Y
*
) (by Observation 3). Therefore, L
(
X
,
Y
*
)
=
K
(
X
,
Y
*
). Let H be a V-subset of (
X
⊗
π
Y
)
*
≃
L
(
X
,
Y
*
)
=
K
(
X
,
Y
*
) and let (
T
n
) be a sequence in H.
Letx
∈
X. We show that (
T
n
(
x
)
) is a V-subset of Y
*. Suppose that ∑
‍
y
n is wuc in Y. If T
∈
K
(
X
,
Y
*
), ∑
‍
|
〈
T
,
x
⊗
y
n
〉
|
=
∑
‍
|
〈
x
,
T
*
(
y
n
)
〉
|
<
∞, since ∑
‍
T
*
(
y
n
) is unconditionally convergent. Then ∑
‍
x
⊗
y
n is wuc in X
⊗
π
Y. Since (
T
n
) is a V-set,
(5)
|
〈
T
n
,
x
⊗
y
n
〉
|
=
|
〈
T
n
(
x
)
,
y
n
〉
|
⟶
0
.
Therefore, (
T
n
(
x
)
) is a V-subset of Y
* and thus is weakly precompact.
Lety
*
*
∈
Y
*
*. We show that (
T
n
*
(
y
*
*
)
) is a V-subset of X
*. Let ∑
‍
x
n be wuc in X. For n
∈
N,
(6)
|
〈
T
n
*
(
y
*
*
)
,
x
n
〉
|
=
|
〈
y
*
*
,
T
n
(
x
n
)
〉
|
.
It is enough to show that(
T
n
(
x
n
)
) is weakly null. Suppose that ∑
‍
y
n is wuc in Y. By Lemma 6, ∑
‍
x
n
⊗
y
n is wuc in X
⊗
π
Y. Since (
T
n
) is a V-set,
(7)
|
〈
T
n
,
x
n
⊗
y
n
〉
|
=
|
〈
T
n
(
x
n
)
,
y
n
〉
|
⟶
0
.
Therefore, (
T
n
(
x
n
)
) is a V-subset of Y
* and thus is weakly precompact. We can assume that (
T
n
(
x
n
)
) is weakly Cauchy. If y
∈
Y,
(8)
|
〈
T
n
,
x
n
⊗
y
〉
|
=
|
〈
T
n
(
x
n
)
,
y
〉
|
⟶
0
,
since (
T
n
) is a V-set and ∑
‍
x
n
⊗
y is wuc in X
⊗
π
Y. Hence (
T
n
(
x
n
)
) is w
*-null. Since (
T
n
(
x
n
)
) is also weakly Cauchy, (
T
n
(
x
n
)
) is weakly null. Then (
T
n
*
(
y
*
*
)
) is a V-subset of X
* and thus is relatively weakly compact. By Theorem 1, H is weakly precompact.
(ii) IfW
(
X
,
Y
*
)
=
K
(
X
,
Y
*
), then W
(
Y
,
X
*
)
=
K
(
Y
,
X
*
) (by Observation 2). By (i), Y
⊗
π
X, thus X
⊗
π
Y has property (
w
V
).Theorem 8 (Theorem 2, [15]).
Suppose thatX and Y have property (
V
) and W
(
X
,
Y
*
)
=
K
(
X
,
Y
*
). Then X
⊗
π
Y has property (
V
).Proof.
SinceX and Y have property (
V
), L
(
X
,
Y
*
)
=
W
(
X
,
Y
*
) (by Observation 3). Therefore, L
(
X
,
Y
*
)
=
K
(
X
,
Y
*
). Let H be a V-subset of (
X
⊗
π
Y
)
*
≃
L
(
X
,
Y
*
)
=
K
(
X
,
Y
*
) and let (
T
n
) be a sequence in H. Since X and Y have property (
V
), X
* and Y
* are both weakly sequentially complete [1]; thus L
(
X
,
Y
*
)
=
K
(
X
,
Y
*
) is weakly sequentially complete, by Theorem 3.10 of  [16]. By Theorem 7, we can (and do) assume that (
T
n
) is weakly Cauchy. Then (
T
n
) is weakly convergent, since K
(
X
,
Y
*
) is weakly sequentially complete.Remark 9.
Theorem8 can be proved using Theorems 7 and 4. To see this, let H be a V-subset of (
X
⊗
π
Y
)
*
≃
L
(
X
,
Y
*
) and let (
T
n
) be a sequence in H. The proof of Theorem 7 shows that, for all x
∈
X, (
T
n
(
x
)
) is a V-subset of Y
*, and thus is relatively weakly compact. Similarly, for all y
*
*
∈
Y
*
*, (
T
n
*
(
y
*
*
)
) is a V-subset of X
*and thus is relatively weakly compact. By Theorem 4, (
T
n
) has a weakly convergent subsequence.Corollary 10 (Corollary 4, [15]).
Suppose thatX
=
C
(
K
) and Y
=
l
p, where 2
<
p
<
∞. Then X
⊗
π
Y has property (
V
).Proof.
Note thatX and Y have property (
V
) [1]. If q is the conjugate of p, then 1
<
q
<
2. Every operator T
:
C
(
K
)
→
l
q, 1
<
q
<
2, is compact ([17], page 100). Apply Theorem 8.Corollary 11.
(i) SupposeX has property (
w
V
), Y has property (
V
), and Y
* has the Schur property. Then X
⊗
π
Y has property (
w
V
).
(ii) Suppose thatX has property (
V
), X
* has the Schur property, and Y has property (
w
V
). Then X
⊗
π
Y has property (
w
V
).
(iii) (Corollary 5, [15]) Suppose X and Y have property (
V
). If X
* or Y
* has the Schur property, then X
⊗
π
Y has property (
V
).Proof.
(i) SinceY
* is a Schur space, W
(
X
,
Y
*
)
=
K
(
X
,
Y
*
). Apply Theorem 7.
(ii) LetT
:
X
→
Y
* be weakly compact. Then T
*
|
Y
:
Y
→
X
* is weakly compact and thus is compact, since X
* has the Schur property. Hence T is compact (by Observation 2) and W
(
X
,
Y
*
)
=
K
(
X
,
Y
*
). Apply Theorem 7.
(iii) IfY
* or X
* has the Schur property, then W
(
X
,
Y
*
)
=
K
(
X
,
Y
*
). Apply Theorem 8.The spaceY satisfies the hypothesis (i) when Y has property (
V
), and Y has the DPP and does not contain l
1. Note that Y
* has the Schur property by Theorem 3 of [18]. In particular, we can take Y
=
C
(
K
), with K dispersed (l
1
↪
C
(
K
)).LetJ  be the original James space [19]. Since J is separable and 1-codimensional in J
*
*, all duals of J are separable and l
1 fails to embed in any of them. Moreover, none of these spaces can be weakly sequentially complete. Thus J and its duals have the RDPP and property (
w
V
), but none of these spaces have property (
V
) (since their duals are not weakly sequentially complete).IfX
=
J or its duals and K is dispersed, then the space X
⊗
π
C
(
K
) has property (
w
V
) (by Corollary 11) and does not have property (
V
) (since X does not have property (
V
)).The fact that properties(
V
) and (
w
V
) are inherited by quotients immediately implies the following result, which contains Corollary 3 of  [15].Corollary 12.
Suppose thatW
(
X
*
,
Y
*
)
=
K
(
X
*
,
Y
*
), X
* has property (
w
V
) (resp., property (
V
)), and Y has property (
V
). Then the space N
1
(
X
,
Y
) of all nuclear operators from X to Y has property (
w
V
) (resp., property (
V
)).Proof.
It is known thatN
1
(
X
,
Y
) is a quotient of X
*
⊗
π
Y ([6], page 41). By Theorem 7 (resp., Theorem 8), X
*
⊗
π
Y has property (
w
V
) (resp., (
V
)). Hence N
1
(
X
,
Y
) has property (
w
V
) (resp., property (
V
)).Theorem 13.
Suppose thatW
(
E
,
F
*
)
=
K
(
E
,
F
*
). The following statements are equivalent:(i)
E and F have property (
V
) and l
1
↪
E or l
1
↪
F,
(ii)
E
⊗
π
F has property (
V
).Proof.
(i) ⇒ (ii) Apply Theorem8.
(ii) ⇒ (i) Suppose thatE
⊗
π
F has property (
V
). Then E and F have property (
V
), since property (
V
) is inherited by quotients. We will show that l
1
↪
E or l
1
↪
F. Suppose that l
1
↪
E and l
1
↪
F. Hence L
1
↪
E
* [20]. Also, the Rademacher functions span l
2 inside of L
1; thus l
2
↪
E
*. Similarly l
2
↪
F
*. Then c
0
↪
K
(
E
,
F
*
) ([21, 22]). Thus l
1
↪
c
E
⊗
π
F [14], a contradiction with Observation 1.Observation 4.
Ifl
1
↪
E and l
1
↪
F, then l
2
↪
E
*, l
2
↪
F
*, and c
0
↪
K
(
E
,
F
*
) ([21, 22]). More generally, if l
1
↪
E, l
p
↪
F
*, and p
≥
2, then c
0
↪
K
(
E
,
F
*
) ([21, 22]). Thus l
1
↪
c
E
⊗
π
F [14], and E
⊗
π
F does not have property (
w
V
) (by Observation 1).Corollary 14.
Suppose thatW
(
E
,
F
*
)
=
K
(
E
,
F
*
) and E has property (
V
). The following statements are equivalent:(i)
F has property (
w
V
) and l
1
↪
E or l
1
↪
F,
(ii)
E
⊗
π
F has property (
w
V
).Proof.
(i) ⇒ (ii) Apply Theorem7.
(ii) ⇒ (i) IfE
⊗
π
F has property (
w
V
), then F has property (
w
V
), since property (
w
V
) is inherited by quotients. Apply Observation 4.The next result contains Corollary 6 of [15].Corollary 15.
Suppose thatE and F have the D
P
P. The following statements are equivalent:(i)
E and F have property (
V
) (resp., property (
w
V
)) and l
1
↪
E or l
1
↪
F,
(ii)
E
⊗
π
F has property (
V
) (resp., property (
w
V
)).Proof.
(i) ⇒ (ii) Suppose thatE and F have the DPP. Without loss of generality suppose l
1
↪
E. Then E
* has the Schur property [18]. Suppose that E and F have property (
V
). Apply Corollary 11(iii). Now suppose that E and F have property (
w
V
). Since E has property (
w
V
) and E
* is weakly sequentially complete, E has property (
V
). Apply Corollary 11(ii). (ii) ⇒ (i) by Observation 4.By Corollary15, the space C
(
K
1
)
⊗
π
C
(
K
2
) has property (
V
) if and only if either K
1 or K
2 is dispersed. The spaces A and H
∞ have the DPP and property (
V
) and contain copies of l
1 ([23–26]). Let E, F be A or H
∞. Then E
⊗
π
F does not have property (
w
V
) (by Observation 4).Next we present some results about the necessity of the conditionL
(
E
,
F
*
)
=
K
(
E
,
F
*
).Theorem 16.
SupposeE
⊗
π
F has property (
V
). Then L
(
E
,
F
*
)
=
K
(
E
,
F
*
), provided any one of the following holds.(i)
(Theorem 7, [15]) Either E
* or F
* has the metric approximation property.
(ii)
IfT
:
E
→
F
* is an operator which is not compact, then there is a sequence (
T
n
) in K
(
E
,
F
*
) such that, for each x
∈
E, the series ∑
‍
T
n
(
x
) converges unconditionally to T
(
x
).
(iii)
EitherE
* or F
* has an (u.c.e.i.).
(iv)
(Theorem 8, [15]) F
* is complemented in a Banach space Z which has an unconditional Schauder decomposition (
Z
n
), and W
(
E
,
Z
n
)
=
K
(
E
,
Z
n
) for all n.
(v)
(Theorem 7, [15]) E
* or F
* has the compact approximation property and is a closed subspace of a Banach space Z possessing an (u.c.e.i.).
(vi)
(Theorem 7, [15]) Either E or F has an (u.c.e.i.).Proof.
SupposeE
⊗
π
F has property (
V
). Then (
E
⊗
π
F
)
*
≃
L
(
E
,
F
*
) is weakly sequentially complete [1].
(i) IfE
* or F
* has the metric approximation property and L
(
E
,
F
*
) is weakly sequentially complete, then W
(
E
,
F
*
)
=
K
(
E
,
F
*
) by Corollary 2.4 of [27].
(ii) LetT
:
E
→
F
* be a noncompact operator. Let (
T
n
) be a sequence as in the hypothesis. By the uniform boundedness principle, {
∑
n
∈
A
‍
T
n
:
A
⊆
N
,
A
finite
} is bounded in K
(
E
,
F
*
). Then ∑
‍
T
n is wuc and not unconditionally convergent (since T is noncompact). Hence c
0
↪
K
(
E
,
F
*
) [14] and we have a contradiction.
(iii) Suppose thatF
* has an (u.c.e.i.) (
A
n
). Then A
n
:
F
*
→
F
* is compact for each n and ∑
‍
A
n
(
y
) converges unconditionally to y, for each y
∈
F
*. Let T
:
E
→
F
* be a noncompact operator. Hence ∑
‍
A
n
T
(
x
) converges unconditionally to T
(
x
) for each x
∈
E and A
n
T
∈
K
(
E
,
F
*
). Then c
0
↪
K
(
E
,
F
*
) (by (ii)), and we have a contradiction.
Similarly, ifE
* has an (u.c.e.i.) and L
(
E
,
F
*
)
≠
K
(
E
,
F
*
), then c
0
↪
K
(
F
,
E
*
).
(iv) LetT
:
E
→
F
* be a weakly compact and noncompact operator, P
n
:
Z
→
Z
n, P
n
(
∑
‍
z
i
)
=
z
n, and let P be the projection of Z onto F
*. Define T
n
:
E
→
F
* by T
n
(
x
)
=
P
P
n
T
(
x
), x
∈
E, n
∈
N. Note that P
n
T is compact since W
(
E
,
Z
n
)
=
K
(
E
,
Z
n
). Then T
n is compact for each n. Since, for each z
∈
Z, ∑
‍
P
n
(
z
) converges unconditionally to z, ∑
‍
T
n
(
x
) converges unconditionally to T
(
x
) for each x
∈
E. By (ii), c
0
↪
K
(
E
,
F
*
), and we have a contradiction.
(v) IfW
(
E
,
F
*
)
≠
K
(
E
,
F
*
), then c
0
↪
K
(
E
,
F
*
) (by Corollary 2.5 of [7]), a contradiction.
(vi) IfL
(
E
,
F
*
)
≠
K
(
E
,
F
*
), then c
0
↪
K
(
E
,
F
*
) (by Theorem 6 of [11]), a contradiction.
Thus,W
(
E
,
F
*
)
=
K
(
E
,
F
*
). Since E and F have property (
V
), L
(
E
,
F
*
)
=
W
(
E
,
F
*
) (by Observation 3). Hence L
(
E
,
F
*
)
=
K
(
E
,
F
*
).The proof of Theorem16 and Observation 4 show that if one of the hypotheses (ii)–(vi) holds and E
⊗
π
F has property (
w
V
), then W
(
E
,
F
*
)
=
K
(
E
,
F
*
). Hence l
p
⊗
π
l
q, where 1
<
p
≤
q
′
<
∞ and q and q
′ are conjugates, does not have property (
w
V
), since the natural inclusion map i
:
l
p
→
l
q
′ is weakly compact and not compact. The space C
(
K
)
⊗
π
l
p does not have property (
w
V
), if K is not dispersed and 1
<
p
≤
2 (by Observation 4, since l
1
↪
C
(
K
) and l
2
↪
l
p
*).By Theorem 6 of [11], c
0
↪
L
(
l
2
,
l
2
); thus l
1
↪
c
l
2
⊗
π
l
2. Since l
2
↪
c
L
p
[
0,1
], for 1
<
p
<
∞ ([6]), l
2
⊗
π
l
2
↪
c
L
p
1
[
0,1
]
⊗
π
L
p
2
[
0,1
] (by Proposition 2.4 of [6]). Thus l
1
↪
c
L
p
1
[
0,1
]
⊗
π
L
p
2
[
0,1
], for 1
<
p
1
,
p
2
<
∞. Hence L
p
1
[
0,1
]
⊗
π
L
p
2
[
0,1
], with 1
<
p
1
,
p
2
<
∞, does not have property (
w
V
) (by Observation 1).Theorem 17.
Assume one of the following holds.(i)
E is an L
∞-space and F
* is a subspace of an L
1-space.
(ii)
E
=
C
(
K
), K is a compact Hausdorff space, and F
* is a space with cotype 2.
(iii)
E
* or F
* has the Schur property.
(iv)
E has the D
P
P and l
1
↪
F.
(v)
E and F have the D
P
P.If E
⊗
π
F has property (
w
V
), then L
(
E
,
F
*
)
=
K
(
E
,
F
*
).Proof.
SupposeE
⊗
π
F has property (
w
V
). Then E and F have property (
w
V
).
Suppose (i) or (ii) holds. It is known that any operatorT
:
E
→
F
* is 2-absolutely summing ([28]); hence it factorizes through a Hilbert space. If L
(
E
,
F
*
)
≠
K
(
E
,
F
*
), then c
0
↪
K
(
E
,
F
*
) (by Remark 3 of [29]), a contradiction with Observation 1.
(iii) Suppose thatE
* has the Schur property. Since E has property (
w
V
) and E
* is weakly sequentially complete, E has property (
V
). By Observation 3, L
(
E
,
F
*
)
=
W
(
E
,
F
*
). Since E
* has the Schur property, W
(
E
,
F
*
)
=
K
(
E
,
F
*
) (see the proof of Corollary 11). Hence L
(
E
,
F
*
)
=
K
(
E
,
F
*
).
Similarly, ifF
* has the Schur property, then L
(
F
,
E
*
)
=
K
(
F
,
E
*
). By Observation 2, L
(
E
,
F
*
)
=
K
(
E
,
F
*
).
(iv) By Observation4, l
1
↪
E. Since E also has the DPP, E
* has the Schur property ([18]). Apply (iii).
(v) Ifl
1
↪
F, then (iv) implies L
(
E
,
F
*
)
=
K
(
E
,
F
*
). If l
1
↪
F, then F
* has the Schur property ([18]); thus L
(
E
,
F
*
)
=
K
(
E
,
F
*
) (by (iii)).



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*Source: 405635-2014-07-01.xml*