# Weak Precompactness in the Space of Vector-Valued Measures of Bounded Variation

Authors: Ioana Ghenciu
Journal: Journal of Function Spaces
 (2015)

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## Abstract

For a Banach spaceX 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).


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## Body

## 1. Introduction

LetX 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 ifl1X*, 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 subsetS 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 subsetA of X (resp., A of X*) is called a V*-subset of X (resp., a V-subset of X*) provided that
(1)lim⁡nsup⁡xn*x:x∈A=0resp.lim⁡nsup⁡x*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 distanced(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]).SupposeΩ 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*).Ifm: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 functionT: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].Letλ 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,(i)
for everyx*∈X*, the map ω→〈ρ(m)(ω),x*〉 is λ-integrable;(ii)
for everyA∈Σ and all x*∈X*,
(4)mA,x*=∫A‍ρmω,x*dλ;(iii)
the mapω→ρ(m)(ω) is λ-integrable and for every A∈Σ,
(5)m(A)=∫A‍ρ(m)(ω)dλ.IfX=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.


## 3. Weak Precompactness inM(Ω,X)

LetX 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.Letρ 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 [1, 6]).
LetA 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 [5, 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,(a)
forω∈C, (ρ(mn′)(ω)) is weakly Cauchy;(b)
forω∈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.
LetA 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 thatA 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,(a)
forω∈C, (ρ(mn′)(ω)) is weakly Cauchy in X**;(b)
forω∈L, there exists k∈N such that (ρ(mn′)(ω))n≥k is equivalent to (en*).
Ifλ(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.
LetA 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 Theorem3.Corollary 5.
LetA be a subset of W(λ,X).(i)
Suppose that, for a.e.ω, the set A(ω) is weakly precompact. Then A is weakly precompact.(ii)
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.(i)
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.(ii)
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 thatA is a subset of W(λ,X).(i)
Ifl1X**, then A is weakly precompact.(ii)
IfX is reflexive, then A is relatively weakly compact.Proof.
Apply Corollary5.Corollary 7.
LetA 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 Corollary5.Corollary 8.
Suppose thatA is a subset of W(λ,X*). If l1X*, then A is weakly precompact.Proof.
Apply Corollary7.The statements of Corollaries6 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]).Lemma 9 (see [3, Lemma 2.15]; [18, page 227]; [9, Corollary 1.7]).
LetA 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 [3, Lemma 2.12]).
LetA 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) IfK is a V*-set in M(Ω,X), then V(K)={|m|:m∈K} is uniformly countably additive.
(ii) IfK is a weakly precompact subset of M(Ω,X), then V(K)={|m|:m∈K} is uniformly countably additive.Proof.
(i) SupposeK 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 lim⁡n|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) IfK is a weakly precompact set in M(Ω,X), then K is a V*-set in M(Ω,X) [7]. Apply (i).LetB(0) denote the unit ball of M(Ω,X).Theorem 13.
(i) SupposeA 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) SupposeA 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) LetA 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 sup⁡jgj(ω)<∞λ-a.e., and thus Ω=∪N{ω:sup⁡jgj(ω)<N}∪Z, where Z is a set of measure zero.
Letϵ>0. Choose δ>0 from the definition of uniform λ-continuity. Choose a positive integer N so that
(11)λω:sup⁡jgjω>N<δ,
and let E={ω:sup⁡jgj(ω)≤N}.
For eachn, 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 eachn,
(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) SupposeA 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, letA 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).LetB denote the unit ball of M(Ω,X*).Theorem 14.
(i) SupposeA 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) SupposeA 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 Theorem13(i).
(ii) IfA 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) Assumel1X**. 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) AssumeX 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.
IfA is a weakly precompact subset of M(Ω,X), then V(A) is uniformly countably additive, by Lemma 12.
Now supposeA 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).(i)
By Corollary6, Hϵ is weakly precompact, since l1X**. By Theorem 13(ii), A is weakly precompact.(ii)
By Corollary6, 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) Ifl1X**, then M(Ω,X) has property (wV*).
(ii) IfX is reflexive, then M(Ω,X) has property (V*).Proof.
LetA be a V*-subset of M(Ω,X). By Lemma 12, V(A) is uniformly countably additive.(i)
By Corollary15(i), A is weakly precompact.(ii)
By Corollary15(ii), A is relatively weakly compact.Corollary 17.
Supposel1X*. 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 Corollary15, using Theorem 14 and Corollary 8.Corollary 18 (see [21], Theorem 3.12, [8]).
SupposeΩ is a compact Hausdorff space and Y is a Banach space. If l1X* and m↔T:C(Ω,X)→Y is a strongly bounded operator, then T*:Y*→C(Ω,X)* is weakly precompact.Proof.
Suppose thatl1X* 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.
IfΩ is a compact Hausdorff space and l1X*, then C(Ω,X) has property (wV).Proof.
Letm↔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.
Supposel1X. Then the following are equivalent: (i)
c0 is not a quotient of X;(ii)
for any compact Hausdorff spaceΩ 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 l1X, then l1X*, 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) Letm↔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 onC(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 thatX is injective and l1Y*. Then Kw*(X*,Y) has property (wV).
(ii) Suppose thatX is injective and Y is reflexive. Then Kw*(X*,Y) has property (V).Proof.
The spaceKw*(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].(i)
By Corollary19, C(BX*,Y) has property (wV). Hence Kw*(Y*,X) has property (wV).(ii)
By Corollary21, C(BX*,Y) has property (V). Hence Kw*(Y*,X) has property (V).Corollary 23.
(i) Suppose thatZ is an L∞-space and l1Y*. Then K(Z*,Y) has property (wV).
(ii) Suppose thatZ is an L∞-space and Y is reflexive. Then K(Z*,Y) has property (V).Proof.
The spaceK(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.



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*Source: 213679-2015-01-06.xml*