For a Banach space X and a measure space (Ω,Σ), let M(Ω,X) be the space of all X-valued countably additive measures on (Ω,Σ) of bounded variation, with the total variation norm. In this paper we give a characterization of weakly precompact subsets of M(Ω,X).

1. Introduction

Let X be a Banach space, (Ω,Σ) a measure space, and M(Ω,X) the space of all X-valued countably additive measures on (Ω,Σ) of bounded variation (with the total variation norm). For m∈M(Ω,X) we denote by |m| its variation. For λ, a probability measure on (Ω,Σ), we denote W(λ,X)={m∈M(Ω,X):|m|≤λ}.

Ulger [1] and Diestel et al. [2] gave a characterization of weakly compact subsets of L1(μ,X), the Banach space of all X-valued Bochner integrable functions on a probability space (Ω,Σ,μ). In [3] we gave a characterization of weakly precompact subsets of L1(μ,X). Randrianantoanina and Saab [4] gave a characterization of relatively weakly compact subsets of M(Ω,X).

In this paper we use results of Talagrand [5], Ulger [6], and techniques of Randrianantoanina and Saab [4] to characterize weakly precompact subsets of M(Ω,X). The characterization is obtained in two steps. In the first step we characterize the weakly precompact subsets of W(λ,X). We show that a subset A of W(λ,X) is weakly precompact if and only if for any sequence (mn) in A and for any lifting ρ of L∞(λ) there exists a sequence (mn′) with mn′∈co{mi:i≥n} for each n such that, for a.e. ω, the sequence (ρ(mn′)(ω)) is weakly Cauchy. In the second step we show that a subset A of M(Ω,X) is weakly precompact if and only if there is a probability measure λ on (Ω,Σ) such that, for any sequence (mn) in A, there is a sequence (mn′) with mn′∈co{mi:i≥n} for each n such that, for any ϵ>0, there is a positive integer N and a weakly precompact subset Hϵ of NW(λ,X) so that {mn′:n≥1}⊆Hϵ+ϵB(0), where B(0) denotes the unit ball of M(Ω,X).

This paper also contains several corollaries of these results. We show that if l1X*, then a subset A of M(Ω,X*) is weakly precompact if and only if A is bounded and V(A)={|m|:m∈A} is uniformly countably additive.

2. Definitions and Notation

Throughout this paper, X and Y will denote Banach spaces. The unit ball of X will be denoted by BX. The unit basis of l1 will be denoted by (en*), and a continuous linear transformation T:X→Y will be referred to as an operator. The set of all compact operators from X to Y will be denoted by K(X,Y). The set of all w*-w continuous compact operators from X* to Y will be denoted by Kw*(X*,Y).

A bounded subset S of X is said to be weakly precompact provided that every sequence from S has a weakly Cauchy subsequence [5]. For a subset A of X, let co(A) denote the convex hull of A. A series ∑xn in X is said to be weakly unconditionally convergent (wuc) if for every x*∈X* the series ∑|x*(xn)| is convergent. An operator T:X→Y is weakly precompact if T(BX) is weakly precompact and unconditionally converging if it maps weakly unconditionally convergent series to unconditionally convergent ones.

A bounded subset A of X (resp., A of X*) is called a V*-subset of X (resp., a V-subset of X*) provided that
(1)limnsupxn*x:x∈A=0resp.limnsupx*xn:x*∈A=0
for each wuc series ∑xn* in X* (resp., wuc series ∑xn in X ).

In his fundamental paper [7], Pełczyński introduced property (V) and property (V*). The Banach space X has property (V) (resp., (V*)) if every V-subset of X* (resp., V*-subset of X) is relatively weakly compact. The following results were also established in [7]: C(K) spaces have property (V); L1-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; if X has property (V), then X* has property (V*); every weakly Cauchy sequence in X* (resp., in X) is a V-set (resp., a V*-set); consequently, every bounded weakly precompact set in X* (resp., in X) is a V-set (resp., a V*-set). A Banach space X has property weak (V)(wV) if any V-subset of X* is weakly precompact [8]. A Banach space X has property weak (V*)(wV*) if every V*-subset of X is weakly precompact [9].

The Banach-Mazur distance d(E,F) between two isomorphic Banach spaces E and F is defined by inf(TT-1), where the infimum is taken over all isomorphisms T from E onto F. A Banach space E is called an L∞-space (resp., L1-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,l1n)≤λ) for some integer n. Complemented subspaces of C(K) spaces (resp., L1(μ) spaces) are L∞-spaces (resp., L1-spaces) (see [10, Proposition 1.26]). The dual of an L1-space (resp., L∞-space) is an L∞-space (resp., L1-space) (see [10, Proposition 1.27]).

Suppose Ω is a compact Hausdorff space, X and Y are Banach spaces, C(Ω,X) is the Banach space of all continuous X-valued functions (with the supremum norm), and Σ is the σ-algebra of Borel subsets of Ω. It is known from [11] that C(Ω,X)*≃M(Ω,X*).

If m:Σ→L(X,Y**) is a finitely additive vector measure and y*∈Y*, then my*(A)=y*m(A)(x), A∈Σ, x∈A. For each y*∈Y*, my*:Σ→X* is a finitely additive vector measure.

Every continuous linear function T:C(Ω,X)→Y may be represented by a vector measure m:Σ→L(X,Y**) of finite semivariation (see [11, 12], and [13, page 182]) such that
(2)T(f)=∫Ωfdm,f∈C(Ω,X),T=m~(Ω),
and T*(y*)=my*, y*∈Y*, where m~ denotes the semivariation of m. We denote this correspondence m↔T. We note that, for f∈C(Ω,X), ∫Ωfdm∈Y even if m is not L(X,Y)-valued. A representing measure m is called strongly bounded if (m~(Ai))→0 for every decreasing sequence (Ai)→∅ in Σ, and an operator m↔T:C(Ω,X)→Y is called strongly bounded if m is strongly bounded [11].

By Theorem 4.4 of [11], a strongly bounded representing measure takes its values in L(X,Y). It is well known that if T is unconditionally converging, then m is strongly bounded [14].

Let λ be a probability measure on Σ, m∈M(Ω,X) with |m|≤λ, and let ρ be a lifting of L∞(λ) (see [12, 15]). For each x*∈X*, the scalar measure x*∘m has a density (d/dλ)(x*∘m)∈L∞(λ) (see [4, 5]). We define ρ(m)(ω) to be the element of X** defined by
(3)ρmω,x*=ρddλx*∘mω,hhhhω∈Ω,x*∈X*.

It is well known (see [12, sect. 13, Theorem 5, page 269]) that,

for every x*∈X*, the map ω→〈ρ(m)(ω),x*〉 is λ-integrable;

for every A∈Σ and all x*∈X*,
(4)mA,x*=∫Aρmω,x*dλ;

the map ω→ρ(m)(ω) is λ-integrable and for every A∈Σ,
(5)m(A)=∫Aρ(m)(ω)dλ.

If X=E* is a dual space, then we define ρ(m)(ω) to be the element of X=E* defined by
(6)ρmω,x=ρddλ(x∘m)(ω),ω∈Ω,x∈E.

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Let X be a Banach space, (Ω,Σ) a measure space, and M(Ω,X) the space of all X-valued countably additive measures on (Ω,Σ) of bounded variation (with the total variation norm). Following [4], for λ, a probability measure on (Ω,Σ), we denote
(7)Wλ,X=m∈MΩ,X:m≤λ.

Let ρ be a lifting of L∞(λ) [12, 15]. For a subset A of W(λ,X) and ω∈Ω, let A(ω)={ρ(m)(ω):m∈A}.

The following results will be useful in our study.

Lemma 1 (see [<xref ref-type="bibr" rid="B23">1</xref>, <xref ref-type="bibr" rid="B22">6</xref>]).

Let A be a bounded subset of X. Then A is weakly precompact (resp., relatively weakly compact) if and only if for each sequence (xn) in A there is a sequence (yn) so that yn∈co{xi:i≥n} for each n and (yn) is weakly Cauchy (resp., weakly convergent).

Lemma 2 (see [<xref ref-type="bibr" rid="B20">5</xref>, Theorem 14]).

Let (mn) be a sequence in M(Ω,X). Suppose there is a probability measure λ with |mn|≤λ for each n. Let ρ be a lifting of L∞(λ). Then there is a sequence (mn′) with mn′∈co{mi:i≥n} for each n and two measurable sets C and L such that λ(C∪L)=1 and,

for ω∈C, (ρ(mn′)(ω)) is weakly Cauchy;

for ω∈L, there exists k∈N such that (ρ(mn′)(ω))n≥k is equivalent to (en*).

The most general result of this paper is the following theorem.

Theorem 3.

Let A be a subset of W(λ,X). Then A is weakly precompact if and only if for any sequence (mn) in A and for any/some lifting ρ of L∞(λ) there exists a sequence (mn′) with mn′∈co{mi:i≥n} for each n such that, for a.e. ω, the sequence (ρ(mn′)(ω)) is weakly Cauchy in X**.

Proof.

Suppose that A is a weakly precompact subset of W(λ,X) and let (mn) be a sequence in A. By passing to a subsequence, we can assume that (mn) is weakly Cauchy. Let ρ be a lifting of L∞(λ). By Lemma 2, there is a sequence (mn′) with mn′∈co{mi:i≥n} for each n and two measurable sets C and L such that λ(C∪L)=1 and,

for ω∈C, (ρ(mn′)(ω)) is weakly Cauchy in X**;

for ω∈L, there exists k∈N such that (ρ(mn′)(ω))n≥k is equivalent to (en*).

If λ(L)>0, then, by [5, Theorem 15], there exists k∈N such that (mn′)n≥k is equivalent to (en*). Since (mn′) lies in the set co(A), which is weakly precompact (see [16, page 377], [17, page 27]), one obtains a contradiction. Hence λ(L)=0, and for a.e. ω, (ρ(mn′)(ω)) is weakly Cauchy in X**.

Conversely, let (mn) be a sequence in A, and let ρ be a lifting of L∞(λ). Let (mn′) be a sequence with mn′∈co{mi:i≥n} for each n such that, for a.e. ω, the sequence (ρ(mn′)(ω)) is weakly Cauchy in X**. By [5, Theorem 15], (mn′) is weakly Cauchy. By Lemma 1, A is weakly precompact.

Theorem 4.

Let A be a subset of W(λ,X*). Then A is weakly precompact if and only if for any sequence (mn) in A and for any/some lifting ρ of L∞(λ) there exists a sequence (mn′) with mn′∈co{mi:i≥n} for each n such that, for a.e. ω, the sequence (ρ(mn′)(ω)) is weakly Cauchy in X*.

Proof.

The proof is similar to that of Theorem 3.

Corollary 5.

Let A be a subset of W(λ,X).

Suppose that, for a.e. ω, the set A(ω) is weakly precompact. Then A is weakly precompact.

Suppose that, for a.e. ω, the set A(ω) is relatively weakly compact. Then A is relatively weakly compact.

Proof.

Let (mn) be a sequence in A. By Lemma 2, there exist a sequence (mn′) with mn′∈co{mi:i≥n} for each n and two sets C and L in Σ with λ(C∪L)=1, such that conditions (a) and (b) of Lemma 2 are satisfied.

Since, for a.e. ω, the set co(A(ω)) is weakly precompact (see [16, page 377], [17, page 27]) and the sequence (ρ(mn′)(ω)) lies in this set, we have λ(L)=0. Then for a.e. ω, the sequence (ρ(mn′)(ω)) is weakly Cauchy in X**. By Theorem 3, A is weakly precompact.

By (i), for a.e. ω, the sequence (ρ(mn′)(ω)) is weakly Cauchy in X**. Since, for a.e. ω, the set co(A(ω)) is relatively weakly compact (Krein-Smulian’s theorem) and the sequence (ρ(mn′)(ω)) lies in this set, we have that (ρ(mn′)(ω)) is relatively weakly compact. Then for a.e. ω, (ρ(mn′)(ω)) is weakly convergent. By [4, Theorem 1], A is relatively weakly compact.

Corollary 6.

Suppose that A is a subset of W(λ,X).

If l1X**, then A is weakly precompact.

If X is reflexive, then A is relatively weakly compact.

Proof.

Apply Corollary 5.

Corollary 7.

Let A be a subset of W(λ,X*). If the set A(ω) is weakly precompact for a.e. ω, then A is weakly precompact.

Proof.

The proof is similar to that of Corollary 5.

Corollary 8.

Suppose that A is a subset of W(λ,X*). If l1X*, then A is weakly precompact.

Proof.

Apply Corollary 7.

The statements of Corollaries 6 and 8 can be considered just for W(λ,X) (resp., W(λ,X*)).

For the proofs of the following two theorems we will need the following lemmas. The first lemma contains a well-known result due to Grothendieck about relatively weakly compact sets (see [18, page 227]).

Let A be a bounded subset of X. If for any ϵ>0 there exists a weakly precompact (relatively weakly compact, resp., V*) subset Aϵ of X such that A⊆Aϵ+ϵBX, then A is weakly precompact (relatively weakly compact, resp., a V*-set).

Lemma 10 (see [<xref ref-type="bibr" rid="B11">3</xref>, Lemma 2.12]).

Let A be a bounded subset of X. Then A is a V*-set if and only if for any sequence (xn) in A there is a sequence (zn) so that zn∈co{xi:i≥n} for each n and {zn:n≥1} is a V*-set.

Lemma 11.

(i) (see [19, Lemma 3.7]) Let A be a bounded subset of X*. If for any ϵ>0 there exists a V-subset Aϵ of X* such that A⊆Aϵ+ϵBX*, then A is a V-set.

(ii) (see [19, Proposition 3.6]) Let A be a bounded subset of X*. Then A is a V-set if and only if for any sequence (xn*) in A there is a sequence (zn*) so that zn*∈co{xi*:i≥n} for each n and {zn*:n≥1} is a V-set.

Lemma 12.

(i) If K is a V*-set in M(Ω,X), then V(K)={|m|:m∈K} is uniformly countably additive.

(ii) If K is a weakly precompact subset of M(Ω,X), then V(K)={|m|:m∈K} is uniformly countably additive.

Proof.

(i) Suppose K is a V*-set in M(Ω,X). Since each member of K is a countably additive measure on the σ-algebra Σ, the set V(K)={|m|:m∈K} is uniformly countably additive if and only if limn|m|(An)=0 uniformly for m∈K whenever (An) is a pairwise disjoint sequence in Σ.

Let (mn) be a sequence in K. Without loss of generality suppose that |mn|≤1 for all n. Let (An) be a pairwise disjoint sequence in Σ and ϵ>0 such that |mn|(An)>ϵ for all n. For each n∈N, let (Ani)i=1kn be a partition of An and let (xni*)i=1kn be points in BX* such that
(8)∑i=1knxni*,mnAni>ϵ.
Define the X*-valued simple functions sn by sn=∑i=1knχAnixni*. Note that ∫sndmn>ϵ for all n∈N. Define T:M(Ω,X)→l1 by
(9)T(m)=∑n∫sndmen*.
Note that T is a well-defined operator, ∑T*(en) is wuc, and 〈mn,T*(en)〉=∫sndmn>ϵ for each n. Then {mn:n≥1} is not a V*-set. This contradiction concludes the proof.

(ii) If K is a weakly precompact set in M(Ω,X), then K is a V*-set in M(Ω,X) [7]. Apply (i).

Let B(0) denote the unit ball of M(Ω,X).

Theorem 13.

(i) Suppose A is a bounded subset of M(Ω,X) such that V(A)={|m|:m∈A} is uniformly countably additive in M(Ω). Then there is a probability measure λ on (Ω,Σ) such that, for any sequence (mn) in A, there is a sequence (mn′) with mn′∈co{mi:i≥n} for each n such that, for any ϵ>0, there is a positive integer N and a subset Hϵ of NW(λ,X) so that {mn′:n≥1}⊆Hϵ+ϵB(0).

(ii) Suppose A is a bounded subset of M(Ω,X). Then A is weakly precompact (resp., a V*-set) if and only if there is a probability measure λ on (Ω,Σ) such that, for any sequence (mn) in A, there is a sequence (mn′) with mn′∈co{mi:i≥n} for each n such that, for any ϵ>0, there is a positive integer N and a weakly precompact (resp., a V*-set) subset Hϵ of NW(λ,X) so that {mn′:n≥1}⊆Hϵ+ϵB(0).

Proof.

(i) Let A be a bounded subset of M(Ω,X) such that V(A)={|m|:m∈A} is uniformly countably additive in M(Ω). Then V(A) is relatively weakly compact in M(Ω) [18]. Hence there is a probability measure λ on (Ω,Σ) so that V(A) is uniformly λ-continuous [18]; that is, for any ϵ>0, there is δ>0 such that if B∈Σ,λ(B)<δ, then
(10)|m|(B)<ϵ,
for all m∈A.

Let (mn) be a sequence in A. For each n, let fn be the λ-density of |mn|. Since {|mn|:n≥1} is relatively weakly compact in M(Ω), {fn:n≥1} is relatively weakly compact in L1(λ). Choose a subsequence (fjn) of (fj) so that (fjn) converges weakly to some function f∈L1(λ). By Mazur’s theorem, there is a sequence (gn) with gn∈co{fji:i≥n} for each n so that gn-f→0. By taking a subsequence, if necessary, gn(ω)-f(ω)→0, λ-a.e. Therefore supjgj(ω)<∞λ-a.e., and thus Ω=∪N{ω:supjgj(ω)<N}∪Z, where Z is a set of measure zero.

Let ϵ>0. Choose δ>0 from the definition of uniform λ-continuity. Choose a positive integer N so that
(11)λω:supjgjω>N<δ,
and let E={ω:supjgj(ω)≤N}.

For each n, gn∈co{fji:i≥n}. Suppose gn=∑k≥naknfjk, with akn≥0 and ∑k≥nakn=1, where the sums are finite. Let mn′=∑k≥naknmjk. Note that mn′∈co{mi:i≥n} for each n. Define
(12)mn′χE:Σ⟶Xbymn′χE(B)=mn′(E∩B),B∈Σ,
and let Hϵ={mn′χE:n≥1}.

For each n,
(13)mn′=mn′χE+mn′χEc.
For B∈Σ,
(14)mn′χE(B)=mn′(E∩B)≤∫E∩Bgn(ω)dλ≤Nλ(B).
Then Hϵ⊆NW(λ,X). For each n,
(15)mn′χEc=mn′(Ec)≤∑k≥naknmjk(Ec)≤ϵ,
and thus mn′χEc∈ϵB(0) for each n. Therefore {mn′:n≥1}⊆Hϵ+ϵB(0).

(ii) Suppose A is a weakly precompact subset (resp., a V*-subset) of M(Ω,X). By Lemma 12, V(A)={|m|:m∈A} is uniformly countably additive in M(Ω). By (i), there is a probability measure λ on (Ω,Σ) such that, for any sequence (mn) in A, there is a sequence (mn′) with mn′∈co{mi:i≥n} for each n such that, for any ϵ>0, there is a positive integer N and a subset Hϵ of NW(λ,X) so that {mn′:n≥1}⊆Hϵ+ϵB(0). Since {mn′:n≥1}⊆co(A), which is weakly precompact (resp., a V*-set) (see [16, page 377], [17, page 27], resp., [9]), Hϵ is weakly precompact (resp., a V*-set).

Conversely, let A be a bounded subset of M(Ω,X). Choose λ a probability measure as in the statement. Let (mn) be a sequence in A. Let (mn′) be a sequence with mn′∈co{mi:i≥n} for each n such that, for any ϵ>0, there is a positive integer N and a weakly precompact subset (resp., a V*-subset) Hϵ of NW(λ,X) so that {mn′:n≥1}⊆Hϵ+ϵB(0). By Lemma 9, {mn′:n≥1} is weakly precompact (resp., a V*-set). By Lemma 1 (resp., 10), A is weakly precompact (resp., a V*-subset).

Let B denote the unit ball of M(Ω,X*).

Theorem 14.

(i) Suppose A is a bounded subset of M(Ω,X*) such that V(A)={|m|:m∈A} is uniformly countably additive in M(Ω). Then there is a probability measure λ on (Ω,Σ) such that, for any sequence (mn) in A, there is a sequence (mn′) with mn′∈co{mi:i≥n} for each n such that, for any ϵ>0, there is a positive integer N and a subset Hϵ of NW(λ,X*) so that {mn′:n≥1}⊆Hϵ+ϵB.

(ii) Suppose A is a bounded subset of M(Ω,X*). Then A is weakly precompact (resp., a V-subset) if and only if there is a probability measure λ on (Ω,Σ) such that, for any sequence (mn) in A, there is a sequence (mn′) with mn′∈co{mi:i≥n} for each n such that, for any ϵ>0, there is a positive integer N and a weakly precompact subset (resp., a V-subset) Hϵ of NW(λ,X*) so that {mn′:n≥1}⊆Hϵ+ϵB.

Proof.

(i) The proof is similar to the proof of Theorem 13(i).

(ii) If A is a weakly precompact subset of M(Ω,X*), then A is a V-subset of M(Ω,X*) [7]. Hence V(A) is uniformly countably additive in M(Ω) (see [9, Proposition 2.1], [20, Proposition 2]). The remainder of the proof is similar to that of Theorem 13(ii), using Lemma 9 (resp., 11) and Lemma 1 (resp., 11).

Corollary 15.

(i) Assume l1X**. Then a subset A of M(Ω,X) is weakly precompact if and only if A is bounded and V(A) is uniformly countably additive.

(ii) Assume X is reflexive. Then a subset A of M(Ω,X) is relatively weakly compact if and only if A is bounded and V(A) is uniformly countably additive.

Proof.

If A is a weakly precompact subset of M(Ω,X), then V(A) is uniformly countably additive, by Lemma 12.

Now suppose A is a bounded subset of M(Ω,X) and V(A) is uniformly countably additive. By Theorem 13(i), there is a probability measure λ on (Ω,Σ) such that, for any sequence (mn) in A, there is a sequence (mn′) with mn′∈co{mi:i≥n} for each n such that, for any ϵ>0, there is a positive integer N and a subset Hϵ of NW(λ,X) so that {mn′:n≥1}⊆Hϵ+ϵB(0).

By Corollary 6, Hϵ is weakly precompact, since l1X**. By Theorem 13(ii), A is weakly precompact.

By Corollary 6, Hϵ is relatively weakly compact, since X is reflexive. By Lemma 9, {mn′:n≥1} is relatively weakly compact. By Lemma 1, A is relatively weakly compact.

Corollary 16.

(i) If l1X**, then M(Ω,X) has property (wV*).

(ii) If X is reflexive, then M(Ω,X) has property (V*).

Proof.

Let A be a V*-subset of M(Ω,X). By Lemma 12, V(A) is uniformly countably additive.

By Corollary 15(i), A is weakly precompact.

By Corollary 15(ii), A is relatively weakly compact.

Corollary 17.

Suppose l1X*. Then a subset A of M(Ω,X*) is weakly precompact if and only if A is bounded and V(A) is uniformly countably additive.

Proof.

The proof is similar to that of Corollary 15, using Theorem 14 and Corollary 8.

Suppose Ω is a compact Hausdorff space and Y is a Banach space. If l1X* and m↔T:C(Ω,X)→Y is a strongly bounded operator, then T*:Y*→C(Ω,X)* is weakly precompact.

Proof.

Suppose that l1X* and m↔T:C(Ω,X)→Y is a strongly bounded operator. We claim that T* is weakly precompact. Recall that T* takes values in M(Ω,X*) and that T*(y*)=my*:Σ→X*. Let (yn*) be a sequence in BY* and let mn=myn*=T*(yn*) for each n∈N. Without loss of generality suppose that |mn|≤1 for each n. Since m is strongly bounded, {|mn|:n≥1} is uniformly countably additive (see [11, Lemma 3.1]). By Corollary 17, {mn:n≥1} is weakly precompact. Hence T* is weakly precompact.

Corollary 19.

If Ω is a compact Hausdorff space and l1X*, then C(Ω,X) has property (wV).

Proof.

Let m↔T:C(Ω,X)→Y be an unconditionally converging operator. Then T is strongly bounded [14]. By Corollary 18, T* is weakly precompact. Then C(Ω,X) has property (wV) [8].

Corollary 20.

Suppose l1X. Then the following are equivalent:

c0 is not a quotient of X;

for any compact Hausdorff space Ω and any Banach space Y, an operator m↔T:C(Ω,X)→Y has weakly precompact adjoint whenever m is strongly bounded and m(A)*:Y*→X* is weakly precompact for every A∈Σ.

Proof.

(i)⇒(ii) If c0 is not a quotient of X and l1X, then l1X*, by [21, Proposition 3.8]. Apply Corollary 18.

(ii)⇒(i) Suppose L:X→c0 is a surjection. By [21, Theorem 2.4], there is a compact space Δ and a continuous linear surjection m↔T:C(Δ,X)→c0 so that m is strongly bounded and m(A):X→c0 is compact for all A∈Σ. Since T is a surjection onto c0, T* is an isomorphism on l1, and thus T* is not weakly precompact.

Corollary 21.

(i) [6, 11] If X is reflexive, then every strongly bounded operator T:C(Ω,X)→Y is weakly compact.

(ii) [7] If X is reflexive, then C(Ω,X) has property (V).

Proof.

(i) Let m↔T:C(Ω,X)→Y be a strongly bounded operator. Let (yn*) be a sequence in BY* and mn=T*(yn*), n∈N. By Corollary 18, {mn:n≥1} is weakly precompact in M(Ω,X*). Since X* is weakly sequentially complete, M(Ω,X*) is weakly sequentially complete [5]. Hence {mn:n≥1} is relatively weakly compact. Then T* is weakly compact. Hence T is weakly compact.

(ii) Every unconditionally converging operator on C(Ω,X) is strongly bounded [14] and thus weakly compact (by (i)). Then C(Ω,X) has property (V) [7].

A Banach space is injective if it is complemented in any superspace. We recall that property (wV) (resp., property (V)) is stable under quotients.

Corollary 22.

(i) Suppose that X is injective and l1Y*. Then Kw*(X*,Y) has property (wV).

(ii) Suppose that X is injective and Y is reflexive. Then Kw*(X*,Y) has property (V).

Proof.

The space Kw*(X*,Y) is isomorphic to Kw*(Y*,X). Since X is injective, Kw*(Y*,X) is complemented in Kw*(Y*,C(BX*)). Now, Kw*(Y*,C(BX*)) is isomorphic to C(BX*,Y) [22].

By Corollary 19, C(BX*,Y) has property (wV). Hence Kw*(Y*,X) has property (wV).

By Corollary 21, C(BX*,Y) has property (V). Hence Kw*(Y*,X) has property (V).

Corollary 23.

(i) Suppose that Z is an L∞-space and l1Y*. Then K(Z*,Y) has property (wV).

(ii) Suppose that Z is an L∞-space and Y is reflexive. Then K(Z*,Y) has property (V).

Proof.

The space K(Z*,Y) is isomorphic to Kw*(Z***,Y) [22]. Since Z is an L∞-space, Z* is an L1-space [10], and thus Z** is injective [23]. Apply Corollary 22.

Conflict of Interests

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

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