# Operators on Spaces of Bounded Vector-Valued Continuous Functions with Strict Topologies

Authors: Marian Nowak
Journal: Journal of Function Spaces
 (2015)

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## Abstract

LetX be a completely regular Hausdorff space, and let (E,‖·‖E) and (F,‖·‖F) be Banach spaces. Let Cb(X,E) be the space of all E-valued bounded, continuous functions defined on X, equipped with the strict topologies βz, where z=σ,∞,p,τ,t. General integral representation theorems of (βz,‖·‖F)-continuous linear operators T:Cb(X,E)→F with respect to the corresponding operator-valued measures are established. Strongly bounded and (βz,‖·‖F)-continuous operators T:Cb(X,E)→F are studied. We extend to “the completely regular setting” some classical results concerning operators on the spaces C(X,E) and Co(X,E), where X is a compact or a locally compact space.


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## Body

## 1. Introduction and Terminology

Throughout the paper let(E,·E) and (F,·F) be real Banach spaces, and let E′ and F′ denote the Banach duals of E and F, respectively. By BF′ and BE we denote the closed unit ball in F′ and E, respectively. By L(E,F) we denote the space of all bounded linear operators U:E→F. Given a locally convex space (L,ξ) by L,ξ′ or Lξ′ we will denote its topological dual. We denote by σ(L,K) the weak topology on L with respect to a dual pair 〈L,K〉.Assume thatX is a completely regular Hausdorff space. Let Cb(X,E) stand for the Banach space of all bounded continuous, E-valued functions on X provided with the uniform norm ·. We write Cb(X) instead of Cb(X,R). By CbX,E′ we denote the Banach dual of Cb(X,E). For f∈Cb(X,E) let f~(t)=f(t)E for t∈X.LetB (resp., Ba) be the algebra (resp., σ-algebra) of Baire sets in X, which is the algebra (resp., σ-algebra) generated by the class Z of all zero sets of functions of Cb(X). By P we denote the family of all cozero sets in X. Let B(B,E) stand for the Banach space of all totally B-measurable functions f:X→E (the uniform limits of sequences of E-valued B-simple functions) provided with the uniform norm ·(see [1, 2]). We will write B(B) instead of B(B,R).Strict topologiesβz on Cb(X) and Cb(X,E) (for z=σ,∞,p,τ,t) play an important role in the topological measure theory (see [3–12] for definitions and more details). Recall that a subset H of Cb(X,E) is said to besolid if f1∈Cb(X,E) and f2∈H with f~1(t)≤f~2(t) for t∈X imply that f1∈H. Then βz are locally convex-solid topologies on CbX,E; that is, they have a local base at 0 consisting of convex and solid sets (see [6, Theorem 8.1], [10, Theorem 5]). We have βt⊂βτ⊂β∞⊂βσ⊂T· and βt⊂βp⊂βσ. For a net (fα) in Cb(X,E), fα→0 for βz if and only if f~α→0 for βz in Cb(X) (see [6, 10]).LetCb(X)⊗E stand for the algebraic tensor product of Cb(X) and E; that is, Cb(X)⊗E is the space of all functions ∑i=1n‍(ui⊗xi), where ui∈Cb(X), xi∈E for i=1,…,n, and (ui⊗xi)(t)=ui(t)xi for t∈X. Then Cb(X)⊗E is dense in (Cb(X,E),βz) for z=∞,τ,t (see [6, 8]). Moreover, Cb(X)⊗E is dense in (Cb(X,E),βσ) if X or E is a D-space (see [6, Theorem 5.2], [13]) and in (Cb(X,E),βp) if X is real-compact (see [10, Theorem 7]).LetCrc(X,E) denote the Banach space of all continuous functions h:X→E such that h(X) is a relatively compact set in E, provided with the uniform norm ∥·∥. Then Cb(X)⊗E⊂Crc(X,E)⊂B(B,E).Linear operators from the spacesCrc(X,E) and Cb(X,E), equipped with the strict topologies βz(z=σ,∞,τ) to a locally convex space (F,ξ), were studied by Katsaras and Liu [14], Aguayo-Garrido, Nova-Yanéz and Sanchez [15, 16], and Khurana [17]. In particular, Katsaras and Liu found an integral representation of weakly compact operators S:Crc(X,E)→F and characterizations of (βz,ξ)-continuous and weakly compact operators S:Crc(X,E)→F for z=σ,τ (see [14, Theorems 3, 4, 5]). Aguayo-Arrido and Nova-Yanéz derived a Riesz representation theorem for (βz,ξ)-continuous and weakly compact operators T:Cb(X,E)→F for z=∞,τ in terms of their representing operator measures (see [15, Theorems 5 and 6]). If X is a locally compact space, continuous operators on Co(X,E) were studied by Dobrakov (see [18]) and Mitter and Young (see [19]).In this paper we develop the theory of continuous linear operators fromCb(X,E), equipped with the strict topologies βz(z=σ,∞,p,τ,t) to a Banach space (F,·F). In particular, we extend to “the completely regular setting” some classical results of Brooks and Lewis (see [20, Theorem 5], [21, Theorem 5.2], [22, Theorem 2.1]) concerning operators on the spaces C(X,E) and Co(X,E), where X is a compact or a locally compact space, respectively. In Section 2, using the device of embedding the space B(B,E) into CrcX,E′′ (the Banach bidual of Crc(X,E)), we state the integral representation of bounded linear operators from Crc(X,E) to F. In Section 3 we derive general Riesz representation theorems for (βz,·F)-continuous linear operators T:Cb(X,E)→F(z=σ,∞,p,τ,t) with respect to the corresponding measures m:B→L(E,F′′) (see Theorems 9 and 14 below). Section 4 is devoted to the study of (βσ,·F)-continuous and strongly bounded operators T:Cb(X,E)→F.


## 2. Integral Representation of Bounded Linear Operators onCrc(X,E)

LetM(X) stand for the Banach lattice of all Baire measures on B, provided with the norm ν=ν(X) (= the total variation of ν). Due to the Alexandrov representation theorem CbX′ can be identified with M(X) through the lattice isomorphism M(X)∋ν↦φν∈CbX′, where φν(u)=∫X‍udν for u∈Cb(X) and φν=ν (see [4, Theorem 5.1]).ByM(X,E′) we denote the set of all finitely additive measures μ:B→E′ with the following properties:(i)
for eachx∈E, the function μx:B→R defined by μx(A)=μ(A)(x) belongs to M(X),(ii)
μ(X)<∞, where μ(A) stands for the variation of μ on A∈B.In view of [23, Theorem 2.5] CrcX,E′ can be identified with M(X,E′) through the linear mapping M(X,E′)∋μ↦Φμ∈CrcX,E′, where Φμ(h)=∫X‍hdμ for h∈Crc(X,E) and Φμ=μ(X). Then one can embed B(B,E) into CrcX,E′′ by the mapping π:B(B,E)→CrcX,E′′, where for g∈B(B,E), (1)π(g)(Φμ):=∫X‍gdμforμ∈M(X,E′).LetiF:F→F′′ denote the canonical embedding; that is, iF(y)(y′)=y′(y) for y∈F, y′∈F′. Moreover, let jF:iF(F)→F stand for the left inverse of iF; that is, jF∘iF=idF.Assume thatS:Crc(X,E)→F is a bounded linear operator. Let (2)S^:=S′′∘π:B(B,E)⟶F′′,where S′:F′→CrcX,E′ and S′′:CrcX,E′′→F′′ denote the conjugate and biconjugate operators of S, respectively. Then we can define a measure m:B→L(E,F′′) (called arepresenting measure of S) by(3)mAx:=S^1A⊗x=S′′∘⊗π1A⊗xiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiforA∈B,x∈E.Then m~(X)<∞, where the semivariation m~(A) of m on A∈B is defined by m~(A):=sup⁡∑‍mAixiF′′, where the supremum is taken over all finite B-partitions (Ai) of A and xi∈BE for each i. For y′∈F′ let us put (4)my′(A)(x):=(m(A)(x))(y′)forA∈B,x∈E.Letmy′(A) stand for the variation of my′ on A. Then (see [1, Section 4, Proposition 5]) (5)m~(A)=sup⁡my′A:y′∈BF′.The following general properties of the operatorS^:B(B,E)→F′′ are well known (see [1, Section 6], [2, Section 1], [13, 24]):(6)S^g=∫X‍gdmforg∈BB,E,S^=m~(X),and for each y′∈F′, (7)S^(g)(y′)=∫X‍gdmy′forg∈B(B,E).ForA∈B let (8)∫A‍gdm:=∫X‍1Agdmforg∈B(B,E).From the general properties of S^ it follows that (9)S^CrcX,E⊂iFF,S(h)=jF(∫X‍hdm)forh∈Crc(X,E).Hence for each y′∈F′ we get (10)y′(S(h))=∫X‍hdmy′forh∈Crc(X,E),and hence my′∈M(X,E′). Moreover, we have(11)S=S′=sup⁡S′y′:y′∈BF′=sup⁡y′∘S:y′∈BF′=sup⁡Φmy′:y′∈BF′=sup⁡my′X:y′∈BF′,and using (5) we get (12)S=m~(X).ByM(X,L(E,F′′)) we will denote the space of all measures m:B→L(E,F′′) such that m~(X)<∞ and my′∈M(X,E′) for each y′∈F′. Thus the representing measure m of S belongs to M(X,L(E,F′′)).For anyx∈E define(13)Sxu:=Su⊗xforu∈CbX,mx(A):=mAxforA∈B.Then Sx:Cb(X)→F is a bounded linear operator. Let χ:B(B)→CbX′′ stand for the canonical embedding; that is, for u∈B(B), (14)χ(u)(φν)=∫X‍udνforν∈M(X).Let (15)S^x:=Sx′′∘χ:B(B)⟶F′′.Then (16)S^xCbX⊂iFF,Sxu=jFS^xuforu∈CbX.The following lemma will be useful.Lemma 1.
LetS:Crc(X,E)→F be a bounded linear operator. Then S′′π1A⊗x=Sx′′χ1A for any x∈E and A∈B.Proof.
Lety′∈F′. Then for each u∈Cb(X), (17)y′∘Sxu=y′(S(u⊗x))=∫X‍(u⊗x)dmy′=∫X‍udmx,y′=φmx,y′(u).Hence we have (18)Sx′′χ1Ay′=χ(1A)(Sx′(y′))=χ(1A)(y′∘Sx)=χ(1A)(φmx,y′)=∫X‍1Admx,y′=mx,y′(1A)=mx(1A)(y′).On the other hand, for each h∈Crc(X,E),  (y′∘S)(h)=∫X‍hdmy′=Φmy′(h), and hence (19)S′′π1A⊗x=(1A⊗x)(S′(y′))=π(1A⊗x)(y′∘S)=π(1A⊗x)(Φmy′)=Φmy′(1A⊗x)=∫X‍(1A⊗x)dmy′=my′(A)(x)=mx(1A)(y′).It follows that S′′(π(1A⊗x))=Sx′′(χ(1A)), as desired.From Lemma1 for A∈B and x∈E we get (20)mxA:=S^1A⊗x=S′′π1A⊗x=Sx′′χ1A;that is, (21)mxA=S^x1A,S^x(u)=∫X‍udmxforu∈B(B).Now we are ready to prove the following Bartle-Dunford-Schwartz type theorem (see [25, Theorem 5, pages 153-154]).Theorem 2.
LetS:Crc(X,E)→F be a bounded linear operator and let M(X,L(E,F′′)) be its representing measure. Then for each x∈E the following statements are equivalent.(i)
Sx:Cb(X)→F is weakly compact.(ii)
m(A)(x)∈iF(F) for each A∈B and {jF(m(A)(x)):A∈B} is a relatively weakly compact set in F.(iii)
mx:B→F′′ is strongly bounded.Proof.
(i)⇒(ii) Assume that Sx is weakly compact. Then by the Gantmacher theorem Sx′′(CbX′′)⊂iF(F) and Sx′′:CbX′′→F′′ is weakly compact (see [26, Theorem 17.2]). Hence S^x(B(B))⊂iF(F) and S^x:B(B)→F′′ is weakly compact. In view of (21) for each x∈E, mx(A)∈iF(F) for A∈B and mx:B→F′′ is strongly bounded (see [25, Theorem 1, page 148]). It follows that {jF(m(A)(x)):A∈B} is a relatively weakly compact subset of F (see [24, Theorem 7]).
(ii)⇒(iii) It follows from [24, Theorem 7].
(iii)⇒(i) Assume that mx:B→F′′ is strongly bounded. Then by (21) S^x:B(B)→F′′ is weakly compact and in view of (16) we derive that Sx is weakly compact.


## 3. Integral Representation of Continuous Linear Operators onCb(X,E)

The spaces of allσ-additive, u-additive, perfect, τ-additive, and tight members of M(X) will be denoted by Mσ(X), M∞(X), Mp(X), Mτ(X), and Mt(X), respectively (see [3, 4]). Then CbX,βz′={φν:ν∈Mz(X)} for z=σ,∞,p,τ,t.For the integration theory of functionsf∈Cb(X,E) with respect to μ∈Mz(X,E′) we refer the reader to [6, page 197], [5, Definition 3.10], [27, page 375]. For z=σ,∞,p,τ,t let (22)Mz(X,E′):=μ∈MX,E′:μx∈Mz(X)foreachx∈E.Then |μ|∈Mz(X) if μ∈Mz(X,E′) (see [5, Proposition 3.9], [6, Theorem 3.1], [10, Theorem 1]). For Φ∈CbX,E′ let us put, for u∈Cb(X)+, (23)|Φ|(u):=sup⁡Φf:f∈Cb(X,E),f~≤u.It is known that |Φ|:Cb(X)+→R+ is additive and positively homogeneous and can be extended to a linear functional on Cb(X) (denoted by Φ again) by Φ(u)=Φ(u+)-Φ(u-) for u∈Cb(X).Theorem 3.
Assume thatz=σ and Cb(X)⊗E is dense in (Cb(X,E),βσ) (resp., z=∞; z=p and Cb(X)⊗E is dense in (Cb(X,E),βp); z=τ; z=t). Then the following statements hold.(i)
For a linear functionalΦ on Cb(X,E) the following conditions are equivalent.
(a)
Φ is βz-continuous.(b)
There exists a uniqueμ∈Mz(X,E′) such that (24)Φ(f)=Φμ(f)=∫X‍fdμforf∈Cb(X,E).(ii)
Forμ∈Mz(X,E′), |Φμ|(u)=∫X‍ud|μ|=φ|μ|(u) for u∈Cb(X).Proof.
(i) See [6, Theorems 5.3 and 4.2, Corollary 3.9], [5, Theorem 3.13], and [10, Theorem 8].
(ii) See [6, Theorem 2.1].Assume thatM is a subset of Mz(X,E′) and sup⁡μ∈M|μ|(X)<∞, where z=σ,∞,p,τ,t. Then we say that M satisfies the condition (Cz) if we have the following:(1)
forz=σ: sup⁡{|μ|(Zn):μ∈M}→0 whenever Zn↓∅, (Zn)⊂Z;(2)
forz=∞: for every partition of unity (uα)α∈A for X and every ɛ>0 there exists a finite set Aɛ in A such that sup⁡μ∈M∫X‍(1-∑α∈Aɛ‍uα)d|μ|<ɛ;(3)
forz=p: for every continuous function f from X onto a separable metric space Y and every ɛ>0, there is a compact subset K of Y such that sup⁡μ∈M|μ|(X∖f¯1(K))≤ɛ;(4)
forz=τ: sup⁡{|μ|(Zα):μ∈M}→0 whenever Zα↓∅, (Zα)⊂Z;(5)
forz=t: for every ɛ>0 there exists a compact subset K of X such that sup⁡{|μ|(Z):Z∈Z,Z⊂X∖K}≤ɛ for each μ∈M.The following lemmas will be useful.Lemma 4.
Assume thatM is a subset of Mz(X,E′) and sup⁡μ∈M|μ|(X)<∞, where z=σ and Cb(X)⊗E is βσ-dense in Cb(X,E) (resp., z=∞; z=p and Cb(X)⊗E is βp-dense in Cb(X,E); z=τ; z=t). Then the following statements are equivalent.(i)
{Φμ:μ∈M} is βz-equicontinuous.(ii)
{|Φμ|:μ∈M} is βz-equicontinuous.(iii)
{φ|μ|:μ∈M} is βz-equicontinuous.(iv)
The condition(Cz) holds.Proof.
(i)⇔(ii) See [9, Lemma 2].
(ii)⇔(iii) It follows from Theorem 3.
(iii)⇔(iv) See [4, Theorem 11.14] for z=σ; [28, Proposition 3.6] for z=∞; [28, Proposition 2.6] for z=p; [4, Theorem 11.24] for z=τ; and [28, Proposition 1.1] for z=t.Lemma 5.
Assume thatz=σ and Cb(X)⊗E is βσ-dense in Cb(X,E) (resp., z=∞; z=p, and Cb(X)⊗E is βp-dense in Cb(X,E);  z=τ; z=t). Let μ∈Mz(X,E′). Then for A∈B the following statements hold.(i)
A functionalΦA:Crc(X,E)→R defined by ΦA(h)=∫A‍hdμ is βz|Crc(X,E)-continuous and can by uniquely extended to a βz-continuous linear functional ΦA¯:Cb(X,E)→R, and one will write the following: (25)∫A‍fdμ:=ΦA¯(f)forf∈Cb(X,E).(ii)
∫A‍fdμ≤∫A‍f~dμ for f∈Cb(X,E).Proof.
(i) Assume that(hα) is a net in Crc(X,E) such that hα→0 for βz. Then (26)ΦA(hα)=∫A‍hαdμ≤∫A‍h~αdμ≤∫X‍h~αdμ.Since h~α→0 for βz in Cb(X) and |μ|∈Mz(X), we obtain that ΦA(hα)→0; that is, ΦA is βz|Crc(X,E)-continuous. Since Crc(X,E) is dense in (Cb(X,E),βz), ΦA can be uniquely extended to a βz-continuous linear functional ΦA¯:Cb(X,E)→R (see [29, Theorem 2.6]).
(ii) Assume thatf∈Cb(X,E). Choose a net (hα) in Crc(X,E) such that hα→f for βz. Then h~α→f~ for βz in Cb(X). Then (27)∫A‍h~αdμ-∫A‍f~dμ≤∫A‍h~α-f~dμ≤∫X‍h~α-f~dμ,and hence ∫A‍f~d|μ|=limα∫A‍h~αd|μ|. Since ∫A‍fdμ=ΦA¯(f)=limα∫A‍hαdμ, we get (28)∫A‍fdμ=limα∫A‍hαdμ≤limα∫A‍h~αdμ=∫A‍f~dμ.Forz=σ,∞,p,τ,t let us put (29)MzX,LE,F′′:=m∈M(X,L(E,F′′)):my′∈Mz(X,E′)foreachy′∈F′.Lemma 6.
Assume thatz=σ and Cb(X)⊗E is βσ-dense in Cb(X,E) (resp., z=∞;  z=p, and Cb(X)⊗E is βp-dense in Cb(X,E); z=τ; z=t). Assume that m∈Mz(X,L(E,F′′)) and the set {my′:y′∈F′} satisfies the condition (Cz). Then for A∈B the following statements hold.(i)
An operatorSA:Crc(X,E)→F′′ defined by SA(h)=∫A‍hdm is (βz|Crc(X,E),·F′′)-continuous and can be uniquely extended to a (βz,·F′′)-continuous linear operator SA¯:Cb(X,E)→F′′, and one will write the following. (30)∫A‍fdm:=SA¯(f)forf∈Cb(X,E).(ii)
For eachy′∈F′, (∫A‍fdm)(y′)=∫A‍fdmy′ for f∈Cb(X,E).Proof.
(i) In view of Lemma5 the set {φmy′:y′∈BF′} is βz-equicontinuous in Cb(X)βz′. Assume that (hα) is a net in Crc(X,E) such that hα→0 for βz. Let ɛ>0 be given. Then there exists a neighborhood Vɛ of 0 for βz in Cb(X) such that sup⁡y′∈BF′∫X‍udmy′≤ɛ for u∈Vɛ. Since h~α→0 for βz in Cb(X), choose αɛ such that hα∈Vɛ for α≥αɛ. Hence sup⁡y′∈BF′∫X‍h~αd|my′|≤ɛ for α≥αɛ. It follows that, for α≥αɛ and each y′∈BF′, (31)∫A‍hαdmy′=∫A‍hαdmy′≤∫A‍h~αdmy′≤∫X‍h~αdmy′≤ɛ,and hence, (32)SA(hα)F′′=sup⁡SAhαy′:y′∈BF′≤ɛ.This means that SA:Crc(X,E)→F′′ is (βz|CrcX,E,·F′′)-continuous. Since Crc(X,E) is βz-dense in (Cb(X,E),βz),SA possesses a unique (βz,·F′′)-continuous extension SA¯:Cb(X,E)→F′′ (see [29, Theorem 2.6]). Let (33)∫A‍fdm:=SA¯(f)forf∈Cb(X,E).
(ii) Letf∈Cb(X,E). Choose a net (hα) in Crc(X,E) such that hα→f for βz. By Lemma 5 and (7) for y′∈F′ we have (34)∫A‍fdmy′=(limα(∫A‍hαdm))(y′)=limα(∫A‍hαdmy′)(y′)=limα∫A‍hαdmy′=∫A‍fdmy′.Corollary 7.
Assume thatz=σ and Cb(X)⊗E is βσ-dense in Cb(X,E) (resp., z=∞; z=p and Cb(X)⊗E is βp-dense in Cb(X,E); z=τ; z=t). Assume that m∈Mz(X,L(E,F′′)) and the set {my′:y′∈BF′} satisfies the condition (Cz). Then for A∈B the following statements hold: (35)(a)my′A=sup⁡∫A‍hdmy′:h∈CbX⊗E,h≤1=sup⁡∫A‍fdmy′:f∈CbX,E,f≤1.(b)m~A=sup⁡∫A‍hdmF′′:h∈CbX⊗E,h≤1=sup⁡∫A‍fdmF′′:f∈CbX,E,f≤1.In particular, if U∈P, then (36)(c)my′U=sup⁡∫U‍hdmy′:h∈Cb(X)⊗E,h≤1,supph⊂U∫U‍hdmy′=sup⁡∑i=1n‍∫X‍uidmxi,y′,where the supremum is taken over all finite disjoint supported collections {u1,…,un}⊂Cb(X) with ui≤1 and suppui⊂U and {x1,…,xn}⊂BE. One has(37)(d)m~U=sup⁡∫U‍hdmF′′:h∈CbX⊗E,hhhhhhh∫U‍hdmF′′h≤1,supph⊂U=sup⁡∫U‍fdmF′′:f∈CbX,E,hhhhhhh∫U‍fdmF′′f≤1,suppf⊂U.Proof.
LetA∈B and y′∈F′. Then by Lemma 5 for f∈Cb(X,E) with f≤1 we have (38)∫A‍fdmy′≤∫A‍f~dmy′≤my′(A).On the other hand, let ɛ>0 be given. Then there exist a finite B-partition (Ai)i=1n of A and xi∈BE, i=1,…,n, such that (39)my′(A)-ɛ3≤∑i=1n‍mAixiy′=∑i=1n‍mxi,y′Ai.By the regularity of mxi,y′∈Mz(X) for i=1,…,n, we can choose Zi∈Z, Zi⊂Ai such that |mxi,y′|(Ai∖Zi)≤ɛ/3n for i=1,…,n. Choose pairwise disjoint Vi∈P with Zi⊂Vi for i=1,…,n such that mxi,y′(Vi∖Zi)≤ɛ/3n. Then for i=1,…,n we can choose vi∈Cb(X) with 0≤vi≤1X, vi|Zi≡1, and vi|X∖Vi≡0 (see [4, page 115]). Define ho=∑i=1n‍(vi⊗xi). Then ho≤1 and ∫A‍hodmy′=∑i=1n‍∫A‍vidmxi,y′=∑i=1n‍∫Vi∩A‍vidmxi,y′. Hence we get (40)my′A-ɛ3≤∑i=1n‍mxi,y′Ai-∑i=1n‍mxi,y′Zi+∑i=1n‍∫Zi‍vidmxi,y′-∑i=1n‍∫Vi∩A‍vidmxi,y′+∫A‍hodmy′≤∑i=1n‍mxi,y′(Ai∖Zi)+∑i=1n‍mxi,y′Vi∖Zi+∫A‍hodmy′≤ɛ3+ɛ3+∫A‍hodmy′and hence my′(A)≤∫A‍hodmy′+ɛ. Thus the proof of (a) is complete.
In view of (5), (a), and Lemma 6 we get (41)m~A=sup⁡my′A:y′∈BF′=sup⁡∫A‍hdmy′:h∈Cb(X)⊗E,h≤1,y′∈BF′∫A‍hdmy′=sup⁡⁡∫A‍fdmy′:f∈Cb(X,E),f≤1,y′∈BF′∫A‍fdmy′=sup⁡∫A‍hdmF′′:h∈CbX⊗E,h≤1=sup⁡∫A‍fdmF′′:f∈CbX,E,f≤1;that is, (b) holds.
Assume now thatU∈P. Let Ui=Vi∩U∈P for i=1,…,n. Then mxi,y′(Ui∖Zi)≤mxi,y′(Vi∖Zi)≤ɛ/3n for i=1,…,n. For i=1,…,n choose ui∈Cb(X) with 0≤ui≤1X, ui|Zi≡1, and ui|X∖Ui≡0. Let ho=∑i=1n‍(ui⊗xi). Then ho≤1 and suppho⊂U; and hence by (a), my′(U)≤∫U‍hodmy′+ɛ. Note that ∫U‍hodmy′=∑i=1n‍∫X‍uidmxi,y′, where supp ui are pairwise disjoint and supp ui⊂U for i=1,…,n. Thus (c) holds.
Using (c) we easily show that (d) holds. Thus the proof is complete.Definition 8.
LetT:Cb(X,E)→F be a bounded linear operator. Then the measure m∈M(X,L(E,F′′)) defined by (42)mAx:=T|CrcX,E′′∘π1A⊗xiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiforA∈B,x∈Ewill be called a representing measure of T.Now we state general Riesz representation theorems for continuous linear operators onCb(X,E), provided with the strict topologies βz, where z=σ,∞,p,τ,t.Theorem 9.
Assume thatz=σ and Cb(X)⊗E is βσ-dense in Cb(X,E) (resp., z=∞; z=p, and Cb(X)⊗E is βp-dense in Cb(X,E); z=τ; z=t).(I)
LetT:Cb(X,E)→F be a (βz,·F)-continuous linear operator and let m∈MX,LE,F′′ be its representing measure. Then the following statements hold.
(i)
m∈MzX,LE,F′′ and my′:y′∈BF′ satisfies the condition (Cz).(ii)
For eachy′∈F′, y′(T(f))=∫X‍fdmy′ for f∈Cb(X,E).(iii)
For eachf∈Cb(X,E) and A∈B there exists a unique vector in F′′, denoted by ∫A‍fdm, such that ∫A‍fdmy′=∫A‍fdmy′ for each y′∈F′.(iv)
For eachA∈B, the mapping Cb(X,E)∋f↦∫A‍fdm∈F′′ is a (βz,·F′′)-continuous linear operator.(v)
Forf∈Cb(X,E), ∫X‍fdm∈iF(F) and T(f)=jF(∫X‍fdm).(vi)
T=m~(X).(II)
Letm∈MzX,LE,F′′ and let the set my′:y′∈BF′ satisfy the condition (Cz). Then the statements (iii) and (iv) hold and for f∈Cb(X,E), ∫X‍fdm∈iF(F) and the mapping T:Cb(X,E)→F defined by T(f):=jF(∫X‍fdm) is a (βz,·F)-continuous linear operator. Moreover, m coincides with the representing measure of T and the statements (ii) and (vi) hold.Proof.
(I) In view of (10) for each y′∈F′, y′(T(h))=∫X‍hdmy′ for h∈Crc(X,E). By Theorem 3 for each y′∈F′ there exists a unique μy′∘T∈MzX,E′ such that y′∘T(f)=∫X‍fdμy′∘T for f∈Cb(X,E). It follows that, for each y′∈F′, my′=μy′∘T (see [23, Theorem 2.5]) and this means that m∈MzX,LE,F′′. Hence (43)y′(T(f))=∫X‍fdmy′forf∈Cb(X,E).
Sincey′∘T:y′∈BF′ is βz-equicontinuous in CbX,Eβz′, by Lemma 4 the set my′:y′∈BF′ satisfies the condition (Cz). Thus (i) and (ii) hold. In view of Lemma 6, (iii) and (iv) are satisfied.
According to (9) for each h∈Crc(X,E), ∫X‍hdm∈iF(F) and T(h)=jF(∫X‍hdm). Hence by Lemma 6, ∫X‍fdm∈iF(F). Let f∈Cb(X,E). Choose a net (hα) in Crc(X,E) such that hα→f for βz. Hence (44)Tf=limαT(hα)=limαjF∫X‍hαdm=jFlimα∫X‍hαdm=jF∫X‍fdm.Thus (v) holds. Using (v) and Corollary 7 we get ∥T∥=m~(X).
(II) By Lemma6 the statements (iii) and (iv) are satisfied.
Now letf∈Cb(X,E). Choose a net (hα) in Crc(X,E) such that hα→f for βz. Then by Lemma 6, ∫X‍fdm=SX¯(f)=limα∫X‍hαdm∈iF(F) because ∫X‍hαdm∈iF(F), and it follows that T(=jF∘SX¯) is (βz,·F)-continuous.
Letmo∈MX,LE,F′′ stand for the representing measure of T. Note that, for A∈B, x∈E, and y′∈F′ we have (45)moAxy′=T|CrcX,E′′∘π1A⊗xy′=π1A⊗xT|CrcX,E′y′=π1A⊗xy′∘T|CrcX,E=∫X‍(1A⊗x)dmy′=∫X‍1Admx,y′=mAxy′;that is, mo=m. By the first part of the proof (ii) and (vi) hold. Thus the proof is complete.Following [14, 27] by Mσ(Ba) we denote the space of all bounded countably additive, real-valued, regular (with respect to zero sets) measures on Ba.We defineMσ(Ba,E′) to be the set of all measures μ:Ba→E′ such that the following two conditions are satisfied.(i)
For eachx∈E, the function μx:Ba→R, defined by μx(A)=μ(A)(x) for A∈Ba, belongs to Mσ(Ba).(ii)
|μ|(X)<∞, where for each A∈Ba, we define |μ|(A)=sup⁡|∑‍μ(Ai)(xi)|, where the supremum is taken over all finite Ba-partitions (Ai) of A and all finite collections xi∈BE.It is known that if μ∈Mσ(Ba,E′), then |μ|∈Mσ(Ba) (see [27, Lemma 2.1]).The following result will be of importance (see [27, Theorem 2.5]).Theorem 10.
Letμ∈MσX,E′. Then μ possesses a unique extension μ¯∈MσBa,E′ and μ¯(X)=μ(X).Arguing as in the proof of Lemma6 we can obtain the following lemma.Lemma 11.
Assume thatCb(X)⊗E is βσ-dense in Cb(X,E) and μ∈Mσ(X,E′). Then for A∈Ba the following statements hold.(i)
A functionalΦA:Crc(X,E)→R defined by ΦA(h)=∫A‍hdμ¯ is βσ|Crc(X,E)-continuous and can be uniquely extended to a βσ-continuous linear functional ΦA¯:Cb(X,E)→R, and one will write the following: (46)∫A‍fdμ¯:=ΦA¯(f)forf∈Cb(X,E).(ii)
Forf∈Cb(X,E), ∫A‍fdμ¯≤∫A‍f~dμ¯.ByMσ(X,L(E,F)) we will denote the space of all operator measures m:B→L(E,F) such that m~(X)<∞ and my′∈MσX,E′ for each y′∈F′. By Mσ(Ba,L(E,F)) we will denote the space of all operator measures m:Ba→L(E,F) with m~(X)<∞ such that my′∈MσBa,E′ for each y′∈F′.Remark 12.
Note that in view of the Orlicz-Pettis theorem everym∈Mσ(Ba,L(E,F)) is countably additive in the strong operator topology; that is, for each x∈E, the measure mx:Ba→F defined by mx(A):=m(A)(x) for A∈Ba is countably additive. Moreover, in view of [30, Theorem 2] for each x∈E, mx is inner regular by zero sets and outer regular by cozero sets; that is, for each A∈Ba and ɛ>0 there exist Z∈Z with Z⊂A and P∈P with A⊂P such that mx(A∖Z)≤ɛ and mx(P∖A)≤ɛ, (mx(A) denotes the semivariation of mx on A∈Ba).According to [14, Theorem 7] we have the following theorem.Theorem 13.
Assume thatm∈Mσ(X,L(E,F)) and {m(A)(x):A∈B} is a relatively weakly compact subset of F for each x∈E. Then m possesses a unique extension m¯∈Mσ(Ba,L(E,F)) such that m¯~(X)=m~(X).For a linear operatorT:Cb(X,E)→F and x∈E let Tx(u):=T(u⊗x) for u∈Cb(X). For m∈MσB,LE,F′′ and x∈E let mx(A):=m(A)(x) for A∈B.Theorem 14.
Assume thatCb(X)⊗E is βσ-dense in Cb(X,E).(I)
LetT:Cb(X,E)→F be a (βσ,·F)-continuous linear operator such that Tx:Cb(X)→F is weakly compact for each x∈E, and let m∈MX,LE,F′′ be the representing measure of T. Then the following statements hold.
(i)
m∈MσX,LE,F′′ and m~(Zn)→0 whenever Zn↓∅, (Zn)⊂Z.(ii)
m(A)(x)∈iF(F), for each A∈B, x∈E, and the measure mF:B→L(E,F), defined by mF(A)(x):=jF(m(A)(x)) for A∈B, x∈E, belongs to Mσ(X,L(E,F)) and possesses a unique extension m¯∈Mσ(Ba,L(E,F)) with m¯~(X)=m~(X) which is countably additive both in the strong operator topology and in the weak star operator topology. Moreover, m¯y′=my′¯ for y′∈F′.(iii)
For everyf∈Cb(X,E) and A∈Ba there exists a unique vector in F, denoted by ∫A‍fdm¯, such that, for each y′∈F′, y′(∫A‍fdm¯)=∫A‍fdm¯y′.(iv)
For eachA∈Ba, the mapping TA:Cb(X,E)→F defined by TA(f)=∫A‍fdm¯ is a (βσ,·F)-continuous linear operator.(v)
T(f)=TX(f)=∫X‍fdm¯ for f∈Cb(X,E).(II)
Letm∈MσX,LE,F′′ be such that m~(Zn)→0 whenever Zn↓∅,(Zn)⊂Z and for each x∈E, let mx:B→F′′ be strongly bounded. Then the operator T:Cb(X,E)→F defined by T(f)=jF(∫X‍fdm) is (βσ,·F)-continuous and Tx:Cb(X)→F is weakly compact for each x∈E, and the statements(ii)–(v) hold.Proof.
(I) (i) It follows from Theorem9.
(ii) In view of Theorem2  m(A)(x)∈iF(F) for A∈B, x∈E, and {mF(A)(x):A∈B} is a relatively weakly compact in F for each x∈E. Since mF∈Mσ(X,L(E,F)), by Theorem 13  mF possesses a unique extension m¯∈Mσ(Ba,L(E,F)) with m¯~(X)=m~(X). By the Orlicz-Pettis theorem m¯ is countably additive in the strong operator topology. Moreover, since, for each y′∈F′, m¯y′∈Mσ(Ba)=ca(Ba), we obtain that m¯y′∈caBa,E′. This means that m¯:Ba→L(E,F) is countably additive in the weak star operator topology.
Lety′∈F′. Then for A∈B and x∈E we have m¯y′(A)(x)=my′(A)(x), and by Theorem 10, m¯y′=my′¯.
(iii) ForA∈Ba let SA(h):=∫A‍fdm¯ for h∈Crc(X,E). Proceeding as in the proof of Lemma 6 we can show that SA:Crc(X,E)→F is a (βσ|Crc(X,E),·F)-continuous linear operator, and hence SA possesses a unique (βσ,·F)-continuous linear extension TA:Cb(X,E)→F (see [29, Theorem 2.6]). Let us write the following: (47)∫A‍fdm¯:=TA(f)forf∈Cb(X,E).Let f∈Cb(X,E). Choose a net (hα) in Crc(X,E) such that hα→f for βσ. For each y′∈F′, m¯y′=my′¯ (see (i)) and by Lemma 11 we have (48)y′∫A‍fdm¯=y′limα∫A‍hαdm¯=limαy′∫A‍hαdm¯=limα∫A‍hαdm¯y′=limα∫A‍hαdmy′¯=∫A‍fdmy′¯=∫A‍fdm¯y′.
(iv) It follows from the proof of (iii).
(v) Letf∈Cb(X,E). In view of Theorem 9, for each y′∈F′, y′(T(f))=∫X‍fdmy′. On the other hand by (ii) for y′∈F′ we have y′(∫X‍fdm¯)=∫X‍fdm¯y′=∫X‍fdmy′. It follows that T(f)=∫X‍fdm¯.
(II) Sincemy′:y′∈BF′ satisfies the condition (Cσ), by Theorem 9 for f∈Cb(X,E), ∫X‍fdm∈iF(F) and the mapping T:Cb(X,E)→F defined by T(f):=jF(∫X‍fdm) is a (βσ,·F)-continuous linear operator, and m coincides with the representing measure of T. Hence in view of Theorem 2  Tx:Cb(X)→F is a weakly compact operator. Thus by the first part of the proof the statements (ii)–(v) are satisfied.


## 4. Strongly Bounded Operators onCb(X,E)

Definition 15.
A bounded linear operatorT:Cb(X,E)→F is said to bestrongly bounded if its representing measure m∈MX,LE,F′′ is strongly bounded; that is, m~(An)→0 whenever (An) is a pairwise disjoint sequence in B.
Note thatm∈MX,LE,F′′ is strongly bounded if and only if the family {|my′|:y′∈BF′} is uniformly strongly additive.Now we are ready to state our main results that extend some classical results of Lewis (see [20, Theorem 5], [31, Lemma 1]) and Brooks and Lewis (see [22, Theorem 2.1], [21, Theorem 5.2]) concerning operators on the spaces C(X,E) and Co(X,E), where X is a compact or a locally compact space, respectively.Theorem 16.
Assume thatCb(X)⊗E is βσ-dense in Cb(X,E). Let T:Cb(X,E)→F be a (βσ,·F)-continuous linear operator and let m∈MX,LE,F′′ be its representing measure. Then m∈MσX,LE,F′′ and the following statements are equivalent.(i)
T is strongly bounded.(ii)
sup⁡my′¯An:y′∈BF′→0 whenever An↓∅, (An)⊂Ba (here my′¯∈MσBa,E′ denotes the unique extension of my′∈MσX,E′).(iii)
If(An) is a sequence in Ba such that An↓∅, then there exists a nested sequence (Un) in P such that An⊂Un for n∈N and sup⁡{TfF:f∈Cb(X,E), f≤1 and suppf⊂Un}→0.Proof.
In view of Theorem9  m∈MσX,LE,F′′.
(i)⇒(ii) Assume that T is strongly bounded. Since the family my′:y′∈BF′ is uniformly strongly additive, according to [25, Lemma 1, page 26] the family my′¯:y′∈BF′ is uniformly countably additive (see Theorem 16).
(ii)⇒(i) It follows from [25, Lemma 1, page 26].
(ii)⇒(iii) Assume that (ii) holds and (An) is a sequence in Ba such that An↓∅. Then there exists λ∈ca(Ba)+ such that my′¯:y′∈BF′ is uniformly λ-continuous (see [25, Theorem 4, pages 11-12]). Let ɛ>0 be given. Hence there exists δ>0 such that sup⁡my′¯A:y′∈BF′≤ɛ/2 whenever λ(A)≤δ and A∈Ba. Since λ is zero-set regular, there exists a nested sequence (Un) in P so that An⊂Un and λ(Un∖An)≤δ for n∈N. Hence sup⁡my′¯Un∖An:y′∈BF′≤ɛ/2 for n∈N. In view of (ii) there exists nɛ∈N such that sup⁡my′¯An:y′∈BF′≤ɛ/2 for n≥nɛ. Hence sup⁡my′Un:y′∈BF′≤ɛ for n≥nɛ; that is, sup⁡my′Un:y′∈BF′→0.
Letf∈Cb(X,E), ∥f∥≤1, and suppf⊂Un. Then by Theorem 9 we have (49)T(f)F=sup⁡∫X‍fdmy′:y′∈BF′≤sup⁡∫X‍f~dmy′:y′∈BF′≤sup⁡my′Un:y′∈BF′.It follows that sup⁡{T(f)F:f∈Cb(X,E),∥f∥≤1, suppf⊂Un}→0.
(iii)⇒(ii) Assume that (iii) holds and An↓∅, (An)⊂Ba. Then there exists a nested sequence (Un) in P such that An⊂Un for n∈N and (50)sup⁡{T(f)F:f∈Cb(X,E),f≤1,suppfn⊂Un}⟶0.Assume that (ii) does not hold. Then there exist ɛ>0 and nɛ∈N such that sup⁡my′¯Anɛ:y′∈BF′≥ɛ and T(f)F≤1/8ɛ whenever f∈Cb(X,E), f≤1, and suppf⊂Unɛ. It follows that there exists yo′∈BF′ such that my′¯(Anɛ)≥ɛ. Hence there exist a finite Ba-partition (Bi)i=1k of Anɛ and xi∈BE, i=1,…,k, such that (51)|myo′¯|(Anɛ)-ɛ4≤∑i=1k‍myo′¯Bixi=∑i=1k‍myo′¯xiBi.Since myo′¯xi∈Mσ(Ba) is zero-set regular (see [4, page 118]), we can choose Zi∈Z, Zi⊂Bi, such that myo′¯xi(Bi∖Zi)≤ɛ/4k for i=1,…,k. Choose pairwise disjoint Vi∈P with Zi⊂Vi for i=1,…,k such that |mxi,yo′|(Vi∖Zi)≤ɛ/4k. Let Ui=Vi∩Unɛ for i=1,…,k. Then Ui∈P and mxi,yo′(Ui∖Zi)≤ɛ/4k for i=1,…,k. For i=1,…,k choose ui∈Cb(X) such that 0≤ui≤1X, ui|Zi≡0, and ui|X∖Ui≡0 (see [4, page 115]). Let ho=∑i=1k‍(ui⊗xi). Then ho≤1, suppho⊂Unɛ, and (52)∫Unɛ‍hodmyo′=∑i=1k‍∫Ui‍uidmxi,yo′.
Hence we get(53)myo′¯(Anɛ)-ɛ4≤∑i=1k‍myo′¯xiBi-∑i=1k‍myo′¯xiZi+∑i=1k‍∫Zi‍uidmxi,yo′-∑i=1k‍∫Ui‍uidmxi,yo′+∫Unɛ‍hodmyo′≤∑i=1k‍myo′¯xi(Bi∖Zi)+∑k=1k‍mxi,yo′(Ui∖Zi)+∫Unɛ‍hodmyo′≤ɛ4+ɛ4+∫Unɛ‍hodmyo′.Hence (54)∫Unɛ‍hodmyo′≥myo′¯(Anɛ)-34ɛ≥14ɛ,T(ho)F≥yo′Tho=∫X‍hodmyo′=∫Unɛ‍hodmyo′≥14ɛ.Thus we get a contradiction to T(ho)F≤1/8ɛ.
Thus the proof is complete.Theorem 17.
Assume thatCb(X)⊗E is βσ-dense in Cb(X,E). Let T:Cb(X,E)→F be a (βσ,·F)-continuous and strongly bounded operator and let m∈M(X,L(E,F′′)) be its representing measure. Then the following statements hold.(i)
m∈MσX,LE,F′′ and m(A)(x)∈iF(F) for A∈B, x∈E, and the measure mF:B→L(E,F), defined by mF(A)(x):=jF(m(A)(x)) for A∈B, x∈E, belongs to Mσ(X,L(E,F)) and possesses a unique extension m¯∈Mσ(Ba,L(E,F)) with m¯~(X)=m~F(X)=m~(X) which is variationally semiregular; that is, m¯~(An)→0 whenever An↓∅, (An)⊂Ba.(ii)
For everyf∈Cb(X,E) and A∈Ba there exists a unique vector in F, denoted by ∫A‍fdm¯, such that, for each y′∈F′, y′(∫A‍fdm¯)=∫A‍fdm¯y′.(iii)
For eachA∈Ba, ∫A‍fndm¯→0 whenever (fn) is a uniformly bounded sequence in Cb(X,E) such that fn(t)→0 for t∈X.(iv)
T(f)=∫X‍fdm¯ for f∈Cb(X,E).(v)
T(fn)→0 whenever (fn) is a uniformly bounded sequence in Cb(X,E) such that fn(t)→0 for t∈X.Proof.
(i) Note that, forx∈E, mxAF′′≤m~AxE for A∈B. Hence mx:B→F′′ is strongly bounded, and by Theorems 2 and 14  m(A)(x)∈iF(F) and mF possesses a unique extension m¯∈Mσ(Ba,L(E,F)) with m¯~(X)=m~F(X)=m~(X). Since m¯y′=my′¯ for y′∈F′, by Theorem 16 we have m¯~(An)=sup⁡{|m¯y′|(An):y′∈BF′}→0 whenever An↓∅,(An)⊂Ba.
(ii) It follows from Theorem14 because for each x∈E, Tx:Cc(X)→F is weakly compact (see Theorem 2).
(iii) In view of (i) there existsλ∈ca(Ba)+ such that {|m¯y′|:y′∈BF′} is λ-continuous (see [25, Theorem 4, pages 11-12]). Let (fn) be a sequence in Cb(X,E) such that sup⁡n∥fn∥=M<∞ and fn(t)→0 for every t∈X. Let ɛ>0 be given. Then there exists δ>0 such that sup⁡{|m¯y′|(A):y′∈BF′}≤ɛ/2M whenever λ(A)≤δ, A∈Ba. Since f~n∈B(B) for n∈N, by the Egoroff theorem there exists Aδ∈Ba with λ(X∖Aδ)≤δ and sup⁡t∈Aδf~n(t)→0. Choose nɛ∈N such that sup⁡t∈Aδf~n(t)≤ɛ/2m~(X) for n≥nɛ.
LetA∈Ba. Note that m¯y′=my′¯ for y′∈F′. Then by Lemma 11 and (ii), for n≥nɛ and y′∈BF′ we get (55)y′∫A‍fndm¯=∫A‍fndm¯y′≤∫A‍f~ndm¯y′≤∫X‍f~ndm¯y′=∫Aδ‍f~ndm¯y′+∫X∖Aδ‍f~ndm¯y′≤ɛ2m~Xm¯y′(Aδ)+M·m¯y′X∖Aδ≤ɛ2m~Xmy′X+M·ɛ2M≤ɛ2+ɛ2=ɛ.Hence ∫A‍fndm¯F≤ɛ for n≥nɛ, as desired.
(iv) It follows from Theorem14.
(v) It follows from (iii) and (iv).LetL∞(Ba,E) stand for the Banach space of all bounded strongly Ba-measurable functions g:X→E, equipped with the uniform norm ∥·∥. Assume that m:B→L(E,F) with m~(X)<∞ is variationally semiregular. Then every g∈L∞(Ba,E) is m-integrable (see [32, Definition 2, page 523 and Theorem 5, page 524]) and ∫X‍gndm→0 whenever (gn) is a uniformly bounded sequence in L∞(Ba,E) converging pointwise to 0 (see [33, Proposition 2.2]).Recall that a series∑i=1∞‍zi in a Banach space G is calledweakly unconditionally Cauchy (wuc) if, for each z′∈G′, ∑i=1∞‍z′zi<∞. We say that a linear operator T:G→F isunconditionally converging if for every weakly unconditionally Cauchy series ∑i=1∞‍zi in G, the series ∑i=1∞‍T(zi) converges unconditionally in a Banach space F.As an application of Theorem17 we have the following result.Corollary 18.
Assume thatCb(X)⊗E is βσ-dense in Cb(X,E), where E is a separable Banach space which contains no isomorphic copy of co. Let T:Cb(X,E)→F be a (βσ,∥·∥F)-continuous and strongly bounded operator. Then T is unconditionally converging.Proof.
Assume that∑i=1∞‍fi is a wuc series in the Banach space Cb(X,E). Hence ∑i=1∞‍x′fit<∞ for each t∈X and x′∈E′ because δt,x′∈CbX,E′, where δt,x′(f)=x′(f(t)) for f∈Cb(X,E). It follows that ∑i=1∞‍fi(t) is an unconditionally convergent series in E for each t∈X because E contains no isomorphic copy of co (see [34]). Let go(t)=limnSn(t) for t∈X, where Sn(t)=∑i=1n‍fi(t) for t∈X, n∈N. Then sup⁡n∥Sn∥<∞ because ∑i=1∞‍fi is wuc (see [34]) and Sn∈L∞(Ba,E) because E is assumed to be separable (see [2, Theorem 21, page 9]). Hence go∈L∞(Ba,E) (see [2, Theorem 10, page 6]).
Letm∈MσX,LE,F′′ be the representing measure of T and let m¯∈Mσ(Ba,L(E,F)) be a unique extension of mF∈Mσ(B,L(E,F)) (see Theorem 17). Since m¯ is variationally semiregular, in view of [33, Proposition 2.2] we have (56)limn∑i=1n‍T(fi)=limn∫X‍Sndm¯=∫X‍godm¯∈E.Hence ∑i=1∞‍T(fi)=∫X‍godm¯. Finally, if (nj) is any permutation of N, then limn∑j=1n‍fnj(t)=go(t) for t∈X. Then ∑j=1∞‍T(fnj)=∫X‍godm¯, as desired.Remark 19.
A related result to Corollary18 for strongly bounded operators on the space Co(X,E) of E-valued continuous functions vanishing at infinity defined on a locally compact space X was obtained by Brooks and Lewis (see [21, Theorem 5.2]).Recall that a Banach spaceE is said to be a Schur space if every weakly convergent sequence in E is norm convergent.As a consequence of Theorem17 we derive the following Dunford-Pettis type theorem for operators on Cb(X,E).Theorem 20.
Assume thatCb(X)⊗E is βσ-dense in Cb(X,E), where E is a Schur space. Let T:Cb(X,E)→F be a (βσ,∥·∥F)-continuous and strongly bounded operator. Then T(fn)→0 in F whenever (fn) is a σCbX,E,MσX,E′ convergent to 0 sequence in Cb(X,E).Proof.
Assume thatfn→0 for σCbX,E,MσX,E′. Then according to [11, Corollary 5], we obtain that sup⁡nfn<∞ and fn(t)→0 in σE,E′ for each t∈X. It follows that fn(t)E→0 for t∈X because E is supposed to be a Schur space. Using Theorem 17 we derive that T(fn)→0 in F, as desired.



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*Source: 407521-2015-04-06.xml*