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formerly Central European Journal of Mathematics

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Volume 14, Issue 1 (Jan 2016)

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A Riesz representation theory for completely regular Hausdorff spaces and its applications

Marian Nowak
  • Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. Szafrana 4A, 65-516 Zielona Góra, Poland
  • Email:
Published Online: 2016-07-29 | DOI: https://doi.org/10.1515/math-2016-0043

Abstract

Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb(X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · ||F) -continuous operators T : Cb(X, E) → F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (β, || · ||F)-continuous operators T : Cb(X, E) → F. As an application, we study (β, || · ||F)-continuous weakly compact and unconditionally converging operators T : Cb(X, E) → F. In particular, we establish the relationship between these operators and the corresponding Borel operator measures given by the Riesz representation theorem. We obtain that if X is a k-spaceand E is reflexive, then (Cb(X, E), β) has the V property of Pełczynski.

Keywords: Spaces of vector-valued continuous functions; Strict topologies; Operator measures; Strongly bounded operators; Unconditionally converging operators; Weakly compact operators

MSC 2010: 46G10; 46E40; 46A70; 28A32

1 Introduction and terminology

Throughout the paper let .E, || · ||E) and (F, || · ||F) be 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 from E to F. Given a locally convex space (Z, ξ) by (Z, ξ)′ or Zξ we will denote its topological dual. We denote by σ(Z, Zξ) the weak topology on Z with respect to a dual pair 〈Z, Zξ〉. Let us recall that (Z, ξ) is a generalized DF-space, if Z has a countable fundamental family {Bn : n ∈ ℕ} of absolutely convex ξ-bounded subsets and if ξ is the finest locally convex topology on Z which agrees with ξ on each Bn.

Assume that (X, 𝓣) is a completely regular Hausdorff space. By 𝓚(X) (resp. 𝓕(X) we will denote the family of all compact (resp. finite) sets In X. Let Cb(X, E) stand for the space of all bounded continuous functions f : XE. By τc (resp. τs ) we denote the topology on Cb(X, E) of uniform convergence on all K𝓚(X) (resp. on all M ∈ 𝓕(X)). By τu we denote the topology on Cb(X, E) of the uniform norm || · ||. By Cb(X, E)′ we denote the Banach dual of Cb(X, E). For fCb(X, E) we will write f˜(t)=f(t)E for tX.

Let Crc(X, E) denote the subspace of Cb(X, E) consisting of those functions h such that h(X) is relatively compact.

Let Cb(X) ⊗ E stand for the linear space spanned by the set of all functions of the form ux, where uCb(X), xE, and (ux)(t) = u(t)x for tX.

By 𝓑o we denote the σ-algebraof Borel sets In X. By 𝓢(𝓑o, E) we denote the set of all E-valued 𝓑o-simple functions on X. Let B(𝓑o, E) stand for the Banach space of all totally 𝓑o-measurable functions g : XE, equipped with the uniform norm || · || (see [13]). Then we have (see [3, Proposition 1]):

Cb(X)ECrc(X,E)B(𝓑o,E).

Different classes of strict topologies βz, where z = σ, ∞, p, τ, t, on the spaces Cb(X), Crc(X, E), Cb(X, E) are of importance In the topological measure theory and have been studied In numerous papers (see [415]). Then

τcβtβτββστuandβtβpβσ.

In this paper, we will consider the strict topology βt on Cb(X, E) that is also denoted by β(see [7, 8]) and by βo (see [5, 11]). It is known that if X is locally compact, then the strict βt coincides with the original topology β of Buck(see[4]).

Remark 1.1: From now on, following [4, 7] and [8] we use a symboll β Instead of βt (or βo) and simply call β a strict topology.

Now we collect basic concepts and facts concerning the strict topology β and the weak strict topology δ on Cb(X, E).

Let 𝓥β (resp. 𝓥δ) stand for the set of all bounded functions υ : X → [0, ∞) such that for every ε > 0, {tX : υ(t) ≥ } ∈ 𝓚(X) (resp. {tX : υ(t) ≥ } ∈ 𝓕(X)). The strict topology β (resp. weak strict topology δ) on Cb(X, E) is generated by the family of seminorms: pυ(f) = suptX υ(t)||f(t)||E for fCb(X, E), where υ𝓥β (resp. υ ∈ 𝓥δ). Then β (resp. δ) can be characterized as the finest locally convex topology on Cb(X, E) which coincides with τc (resp. τs) on τu-bounded subsets of Cb(X, E) (see [7, 11, 13]). The topologies δ, β and τu have the same bounded subsets(see [7, Theorem 3.4], [13]). This means that (Cb(X, E), β) and (Cb(X, E), δ) are generalized DF-spaces (see [11], [13, Corollary]); equivalently, β (resp. δ) coincides with the mixed topology γ[τu, τc] (resp. γ[τu, τs]) In the sense of Wiweger (see [16, 17] for more details). We have τsδβτu and τcβ and Cb(X) ⊗ E is β-dense In Cb(X, E) (see [11]). It is known that Cb(X,E)β (resp. Cb(X,E)δ) is equal to the closure of Cb(X,E)τc (resp. Cb(X,E)τs) In the Banach space (Cb(X, E)′, || · ||) (see [17, Proposition 1]). Moreover, a sequence (fn) In Cb(X, E) is convergent to 0 for β (resp. δ) if and only if supn||fn|| < ∞ and fn → 0 for τc (resp. for τs) (see [17, Proposition 1]). Note that δ = β if and only if 𝓚(X) ⊂ 𝓕(X) (see [12, Proposition 3.2]).

If X is a locally compact Hausdorff space, by Co(X, E) (resp. Cc(X, E)) we denote the space of all continuous functions f : XE vanishing at Infinity (resp. with a compact support), equipped with the uniform norm.

For X being a compact Hausdorff space (resp. a locally compact Hausdorff space) and E, F being locally convex spaces (in particular, Banach spaces), the problem of an Integral representation of bounded linear operators from C(X, E) (resp. Co(X, E) and Cc(X, E)) to F In terms of their representing measures m : 𝓑o → 𝓛(E, F″) has been studied by Foias and Singer [18], Dinculeanu [1, 2], Goodrich [19], Dobrakov [20] and Shuchat [3]. Different classes of operators on C(X, E) have been studied intensively; see [1, 2], [2032]. The study of the relationship between operators and their representing operator measures is a central problem in the theory.

Linear operators from the spaces Crc(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 ([33]), Aguayo-Garrido and Nova-Yanéz (see [34, 35]). 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 [33]). 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 Baire measures (see [35, Theorems 5 and 6]).

In [36] we have developed a general theory of continuous linear operators from Cb(X, E), equipped with the strict topologies βz (where z = σ, ∞, p, τ, t) to a Banach space F. Next, in [37] In case Cb(X)E is assumed to be βσ-dense In Cb(X, E) (for example, if X has a σ-compact dense subset; resp. X is a D-space; resp. E is a D-space), we have studied different classes of (βσ, || · ||F)-continuous operators T : Cb(X, E) → F in terms of their representing Baire operator measures.

The main aim of this paper is to develop a Riesz Integral representation theory for (β, || · ||F)-continuous operators T : Cb(X, E) → F, where X is supposed to be a completely regular Hausdorff space. As an application, we extend to "the setting of completely regular spaces (in particular, k-spaces)" the classical results concerning operators on the spaces C(X, E) and Co(X, E), where X is a compact or a locally compact Hausdorff space. In Section ∈ we develop the duality theory of Cb(X, E), equipped with the strict topologies β and δ. In Section 3 using the device of embedding the space B(𝓑o, E) into Cb(X,E)β (= the bidual of (Cb(X, E), β)), the properties that Crc(X, E) ⊂ B(𝓑o, E) and Crc(X, E) is β-dense in Cb(X, E), we derive the integral representation of (β, || · ||F)-continuous operators T : Cb(X, E) → F by the Riemann-Stieltjes type integrals with respect to the representing measures m : 𝓑o → 𝓛(E, F") (see Theorem 3.2 below). In Section 4 using the Dieudonné-Grothendieck type criterion for relative weak compactness in the Banach space M(X) for X being a k-space (see Theorem 3.5), we characterize strongly bounded operators T : Cb(X, E) → F (see Theorems 4.1 and 4.3 below). In Section 5 we establish a Radon-type extension theorem for strongly bounded operators T : Cb(X, E) → F (see Theorem 5.2 below). In Section 6 we extend and generalize some classical result of Brooks and Lewis [26] concerning weakly compact operators on spaces of continuous vector-valued functions (see Theorem 6.2). In Section 7 we study unconditionally converging operators T : Cb(X, E) → F. As an application, we derive that if X is a k-space and E is reflexive, then (Cb(X, E), β) has the V property of Pełczynski (see Corollary 7.9 below).

Remark 1.2: In the present paper we study the Riemann-Stieltjes integration of bounded continuous functions with respect to Borel operator measures. This approach is very natural and has been used by many authors (see [9-11, 14, 1, 33]) and is different from the concept of integration developed by the present author in [36] and [37], where the integral of bounded continuous functions is defined as a βz-continuous extension of the so-called immediate integral of [1], [2, Sections G-H] (see [36, Lemma 6 and Theorem 9]).

2 Duality of Cb(X, E) with the strict topologies

Recall that a countably additive scalar measure ν on 𝓑o is said to be a regular if its variation | ν | : 𝓑o → ℝ+ is regular, i.e., for each A𝓑o,

|ν|(A)=sup{|ν|(K):KK(X),KA}=inf{|ν|(O):O𝓣,OA}.

By M(X) we denote the space of all countably additive regular scalar Borel measures.

Let M(X, E′) denote the space of all countably additive measures μ : 𝓑oE′ of bounded variation (|μ|(X) < ∞) such that for each xE, μxM(X), where μx(A) := μ(A)(x) for A ∈ 𝓑o. Then |μ| ∈ M(X) (see [8, Lemma 2.3]).

Assume that m : 𝓑o → 𝓛(E, F) is a vector measure. Then the semivariation m˜ of m on A𝓑o is defined by m˜(A) : = sup || ∑m(Ai)(xi)||F, where the supremum is taken over all finite 𝓑o-partitions (Ai) of A and xiBF for each i. For each y′F′ let

my(A)(x):=y(m(A)(x))forA𝓑o,xE.

Then my′ : 𝓑oE′ and for A𝓑o, we have (see [1, § 4, Proposition 5])

m˜(A)=sup{|my|(A):yBF}.

By M(X, 𝓛(E, F)) we denote the space of all measures m : 𝓑o → 𝓛(E, F) such that m˜(X) < ∞ and for each y′F′, my′M(X, E′).

Definition 2.1: We say that mM(X, 𝓛(E, F)) has a regular semivariation if for every A𝓑o and ∊ > 0 there exist K𝓚(X) and O ∈ 𝓣 such that KAO and m˜(O \ K) ≤ .

For A𝓑o let 𝒟A denote the collection of all 𝓐 = {A1, …, An; t1, …, tn}, where (Ai)i=1n is a 𝓑o-partition of A and tiAi for i = 1, …, n. For 𝓐1, 𝓐2𝓓A define 𝓐1𝓐2 if each set which appears in 𝓐1 is contained in some 𝓐2. in this way 𝓓A becomes a directed set.

Assume that mM(X, 𝓛(E, F)). Then for fCb(X, E) and 𝓐 = {𝓐1,..., 𝓐n; t1,..., tn} ∈ 𝓓A we will write 𝓢𝓐(f):=i=1nm(Ai)(f(ti)). Following [19, Definition 2], we have a definition.

Definition 2.2: We say that fCb(X, E) is m-integrable over A𝓑o provided there exists yAF such that given ∊ > 0 there exists a finite 𝓑o-partition 𝓟 of A such that yAi=1nm(Ai)(f(ti))Fεif {A1, …, An} is any 𝓑o-partition of A refining 𝓟 and {t1, …, tn} is any choice of points of A such that tiAi for i = 1, …, n. Then ∫A f dm := yA will be called a Riemann-Stieltjes integral of f with respect to m over A𝓑o.

Theorem 2.3: Assume that mM(X, 𝓛(E, F)) has the regular semivariation. Then the following statements hold:

  1. Every fCb(X, E) is m-integrable over all A𝓑o.

  2. For each fCb(X, E) and A𝓑o, y′(∫A f dm) = ∫A f dm y′ for y′ ∈ F′:

Proof. (i) Let fCb(X, E) and A𝓑o. We will show that f is m-integrable over A. Indeed, let η > 0 be given, and ε=η4m˜(X). Then there exists K𝓚(X) with KA such that supyBF|my|(A\K)η8f.For each tA let Wt = {sX : ||f(s) – f(t)||E < ∊}. Hence there exists a set {t1, …, tn} in A such that Ki=1nWti Let V=i=1nVti, where Vti = WtiA for i = 1, …, n. Then V𝓑o and supyBF|my|(A\V)η8f. Define A1=Vt1,Ai=Vti\j=1i1Vtj for i = 2,...,n and An+1 = A \ V. Then V=i=1nAi and ti ∈ 𝓓A, for i = 1,..., n and let tn+1An+1. Hence 𝓐 = {A1,..., An, An+1; t1,..., tn, tn+1) ∈ 𝓓A. Let 𝓐1, 𝓐2𝓓A and 𝓐1 ≥ 𝓐, 𝓐2 ≥ 𝓐, where 𝓐1 = {B1,...,Bp,s1,...,sp} and 𝓐2 = {C1,..., Cq;τ1,…, τq}.

Then we have

S𝓐ε(f)S𝓐1(f)F=i=1n+1m(Ai)(f(ti))j=1pm(Bj)(f(sj))Fi=1njJim(Bj)(f(ti))j=1pm(Bj)(f(sj))F+m(An+1)(f(tn+1))jJn+1m(Bj)(f(sj))Fεi=1njJim(Bj)(f(ti)f(sj)ε)F+supyBF|jJn+1my(Bj)(f(tn+1)f(sj))|εm˜(X)+2fsupyBFjJn+1|my(Bj)(f(tn+1)f(sj)2f)|εm˜(X)+2fsupyBF|my|(An+1)η4+η4=η2.

Similarly, we get S𝓐ε(f)S𝓐2(f)Fη2, and hence

S𝓐1(f)S𝓐2(f)FS𝓐ε(f)S𝓐1(f)F+S𝓐ε(f)S𝓐2(f)Fη.

It follows that (S𝓐(f)) is a Cauchy net in (F, || · ||F) and hence the integral ∫A f dm := lim𝓐 S𝓐(f) exists in F.

Note that for each y′ ∈ F' and 𝓐 = {A1,..., An, t1,..., tn} ∈ 𝓓A, we have y(𝓢𝓐(f))=i=1nmy(Ai)(f(ti)) and ∫A f dmy = lim𝓐 y′(𝓢𝓐(f)). Hence, we get

y∫!Afdm=limAySAf=Afdmy.

In particular, if μM(X, E′), then every fCb(X, E) is μ-integrable over all A ∈ 𝓑o and one can easily show that

|Afdμ|Af˜d|μ|f|μ|(A).(1)

Note that the equation Φμ(f) = ∫X f dμ for fCb(X, E), defines a β-continuous linear functional on Cb(X, E). Indeed, let (fα) be a net in Cb(X, E) such that fα → 0 for β. Then f˜α0 for β in Cb(X) and since |μ| ∈ M(X), we get Xf˜αd|μ|0 (see [10, Lemma 4.2]). By (1) we obtain that Φμ(fα) → 0, as desired.

Conversely, assume that Φ is a β-continuous linear functional on Cb(X, E). Then by [8, Theorem 3.2] there exists a unique μM(X, E′) such that Φ(h) = X h dμ for hCrc(X, E): Since Crc(X, E) is β-dense in Cb(X, E), we obtain that Φ(f) = Φμ(f) =∫X f dμ for fCb(X, E).

Now we can state the following characterization of (β, || · ||F)-continuous functionals on Cb(X, E).

Theorem 2.4: For a linear functional Φ on Cb(X, E) the following statements are equivalent:

  1. Φ is β-continuous.

  2. There exists a unique μM(X, E′) such that Φ(f)=Φμ(f)=XfdμforfCb(X,E).

We will use the following topological result: if (Ki)i=1n are disjoint compact sets in X, then there are disjoint open sets (Oi)i=1n with KiOi for i = 1, …, n, and one can choose υiCb(X) with 0 ≤ υi ≤ 𝟙X such that υi|Ki. ≡ 1 and υi|X\Oi. ≡ 0 (see [38, Theorem 3.1.6 and Theorem 3.1.7]).

The following lemma will be useful.

Lemma 2.5: Assume that μM(X, E′). Then for A𝓑o, |μ|(A)=sup{|Ahdμ|:hCb|(X)E,h=1}=sup{|Afdμ|:fCb(X,E),f1}.(2)Moreover, for O ∈ 𝓣, we have |μ|(O)=sup{|Ohdμ|:hCb(X)E,h=1andsupp  hO}=sup|i=1nX(uixi)dμ|,(3)

where the supremum is taken over finite disjointly supported collections {u1, …, un} in Cb(X) with ||ui|| = 1 and supp uiO and {x1, …, xn} ⊂ BE.

Proof. Let A ∈ 𝓑o. Then for fCb(X, E), ||f|| ≤ 1, by (1) we have |Afdμ|Af˜d|μ||μ|(A). On the other hand, let > 0 be given. Then there exist a finite 𝓑o-partition (Ai)i=1n of A and xiBE, i = 1, …, n such that

|μ|(A)ε3|i=1nμ(Ai)(xi)|=|i=1nμxi(Ai)|.

By the regularity of μxiM(X) for i = 1,...,n, we can choose Ki ∈ 𝓚(X), KiAi such that |μxi|(Ai\Ki)ε3n for i = 1, …, n. Choose pairwise disjoint Oi ∈ 𝓣 with KiOi for i = 1, …, n such that |μxi|(Oi\Ki)ε3n. Then for i = 1,...,n we can choose υiCb(X) with 0 ≤ υi ≤ 𝟙X, υi|Ki ≡ 1, υi|X\Oi 0.

Define ho=i=1n(υixi).Then ||ho|| = 1 and

Ahodμ=i=1nAυidμxi=i=1nOiAυidμxi.

Hence we get, |μ|(A)ε3|i=1nμxi(Ai)i=1nμxi(Ki)|+|i=1nKiυidμxii=1nOiAυidμxi|+|Ahodμ|i=1n|μxi|(Ai\Ki)+i=1n|μxi|(Oi\Ki)+|Ahodμ|ε3+ε3+|Ahodμ|

and hence |μ|(A) ≤ | A hod μ| +∊. Thus the proof of (2) is complete.

Assume that O ∈ 𝓣. Let Vi = OiO for i = 1,...,n. Then |μxi|(Vi\Ki)|μxi|(Oi\Ki)ε3n for i = 1,...,n. For i = 1,...,n choose μiCb(X) with 0 ≤ μi, ≤ 𝟙X, μi|Ki. ≡ 1, μi|X\Vi 0. Let h0=i=1n(uixi). Then ||ho|| = 1 and supp hoO, and hence by (2), |μ|(O) ≤ |∫O h0 |+ ∊. Note that Oh0dμ=i=1nXuidμxi, where supp μi are pairwise disjoint and supp μiO for i = 1, …,n. Thus (3) holds. □

From Lemma 2.5 it follows that if μM(X, E′), then

|μ|(X)=Φμ.(4)

Remark 2.6: Since Cb(X,E)β : a closed subset of the Banach space (Cb(X, E)′, || · ||), in view of Theorem 2.4 and (4) we obtain that M(X, E′), equipped with the norm ||μ|| = |μ|(X) is a Banach space. It follows that M(X, E′) is a closed subspace of the Banach space cabv(𝓑o; E′).

Corollary 2.7: For a subset 𝓜 of M(X, E′) the following statements are equivalent:

  1. μ : μ𝓜} is a β-equicontinuous subset of Cb(X,E)β.

  2. supμ∈𝓜|μ|(X) < ∞ and for every ∊ > 0 there exists K ∈ 𝓚(X) such that supμ∈𝓜|μ|(X \ K) ≤ .

  3. supμ∈𝓜|μ|(X) < ∞ and for every ∊ > 0 there exists K𝓚(X) such that supμ∈𝓜|∫X f dμ∊ if fCb(X, E) with ||f|| ≤ 1 and f|K 0.

Proof. (i) ⇔ (ii) See [15, Lemma 2].

(ii) ⇔ (iii) It follows from Lemma 2.5.

Schmets and Zafarani developed the duality theory of Cb(X), equipped with the weak strict topology δ (see [12, Theorem 4.1 and Proposition 4.2]). Now we extend these results to the vector-valued setting.

For tX and x′E′, let δt,x′(f) := x′(f(t)) for fCb(X, E). Then δt,x is a τs-continuous linear functional and ||δt,x|| = ||x′||E. By Theorem 2.4 there exists a unique μt,xM(X, E′) such that

δt,x(f)=Xfdμt,xforfCb(X,E).

Then t, x′|(X) = ||δt,x|| and using Lemma 2.5, we get |μt, x|(X \ {t}) = 0. Hence |μt, x|({t}) = ||x'||E' and μt,x'({t}) = x′. It follows that μt, x' is concentrated on {t}, that is, for A𝓑o,

μt,x(A)=𝟙A(t)xand|μt,x|(A)=𝟙A(t)xE.

Let C(X, E) stand for the space of all continuous functions f : XE, equipped with the topology τs of simple convergence. It is known that (C(X, E), τs)' = Lin{δt, x′ : tX, x'E'}, where δt,x'(f) = x'(f(t)) for fC(X, E) (see [39], [40, Section 1]). Since Cb(X, E) is dense in (C(X, E), τs) (see [41, Theorem 1.5.3]), we get

(Cb(X,E),τs)=Lin{δt,x:tX,xE}.

Lemma 2.8: Let υ𝓥δ. Then for a linear functional Φ on Cb(X, E) the following statements are equivalent:

  1. There exist a > 0 such that |Φ(f)| ≤ apυ(f) for fCb(X, E).

  2. There exist a sequence (tn) in X and (xn)1(E) such that Φ(f)=n=1υ(tn)xn(f(tn))=n=1υ(tn)Xfdμtn,xn.

Proof. (i) ⇒ (ii) Assume that (i) holds. Then there exists a sequence (tn) in X such that υ(t)=n=𝟙υ(tn)1{tn}(t) for t ∈ X, where υ(tn) → 0. Let

L:={(υ(tn)f(tn)):fCb(X,E)}.

Note that Lco(E), where co(E) is equipped with the supremum norm || · ||. Define a functional Ψυ on L by Ψυ((υ(tn)f(tn))) := Φ(f). Then Ψυ is a || · ||-continuous linear functional and ||Ψυ|| ≤ a. Hence by the Hahn-Banach theorem there exists a || · ||-continuous linear extension Ψυ on co(E) of Ψυ such that ||Ψυ|| = ||Ψυ||. It follows that there exists (xn) in 1(E′) such that Ψυ¯((xn))=n=1xn(xn) for (xn) ∈ co(E) and Ψυ¯=n=1xnEa. Then for fCb(X,E),

Φ(f)=Ψυ¯((υ(tn)f(tn)))=n=1υ(tn)xn(f(tn))=n=1υ(tn)Xfdμtn,xn.

(ii) ⇒ (i) Assume that (ii) holds. Then for fCb(X, E),

|Φ(f)|n=1υ(tn)f(tn)ExnE(n=1xnE)pυ(f).

Definition 2.9: A measure μM(X, E′) is called discrete if for every ∊ > 0 there exists M𝓕(X) such that |μ|(X \ M) ≤ ∊ : By Md(X, E′) we will denote the subspace of M(X, E′) of all discrete measures.

Theorem 2.10: For a linear functional Φ on Cb(X, E) the following statements are equivalent:

  1. Φ : δ-continuous.

  2. There exists a unique μMd(X, E′) such that Φ(f)=Φμ(f)=XfdμforfCb(X,E).

Proof. (i) ⇒ (ii) Assume that ΦCb(X,E)δ. Then there exist υ𝓥δ and a > 0 such that |Φ(f)| ≤ apυ(f) for fCb(X, E). By Lemma 2.8 there is a sequence (tn) in X and (xn)1(E) with n=1xnEa such that Φ(f)=n=1υ(tn)xn(f(tn)) for fCb(X, E). Since

n=1υ(tn)μtn,xn=n=1υ(tn)xnE(n=1xnE)supnυ(tn)<,

we see that μ=n=1υ(tn)μtn,xnM(X,E). It follows that for fCb(X, E),

Xfdμ=n=1υ(tn)Xfdμtn,xn=n=1υ(tn)xn(f(tn))=Φ(f).

Let ∊ > 0 be given. Choose n ∈ ℕ such that n=nε+1υ(tn)xnEε. Hence for M = {t1,..., tn} by Lemma 2.5, we get

|μ|(X\Mε)={|n=nε+1υ(tn)xn(f(tn))|:fCb(X,E),f1,suppfX\Mε}n=nε+1υ(tn)xEε.

This means that μMd(X, E′) and Φ(f) = X f dμ for fCb(X, E).

(ii) ⇒ (i) Assume that (ii) holds. Then we can choose a sequence (Mn) in 𝓕(X) such that MnMn+1 for n ∈ ℕ and |μ|(X \ Mn) ≤ 2–2n. Let υ(t)=n=12n1Mn(t) for tX. Then υ𝓥δ. Assume that fCb(X, E) and pυ(f) = suptXυ(t)||f(t)||E ≤ 1. Then sup{||f(t)||E : tMn+1 \ Mn} ≤ 2n and sup{||f(t)||E : tM1} ≤ 1. Note that |μ|(X\n=1Mn)=0, so X\n=1Mnf˜d|μ|=0. Hence

|Φμ(f)|=|Xfdμ|n=1Mnf˜d|μ|=M1f˜d|μ|+n=1Mn+1\Mnf˜d|μ||μ|(X)+1.

It follows that |Φμ(f) ≤ (|μ|(X) + 1)pυ(f) for fCb(X, E), i.e.,ΦμCb(X,E)δ. □

Corollary 2.11: For a subset 𝓜 of Md(X, E′) the following statements are equivalent:

  1. μ : μ ∈ 𝓜} is a δ-equicontinuous subset of Cb(X,E)δ.

  2. supμ∈𝓜 |μ|(X) < ∞ and for every ∊ > 0 there exists M ∈ 𝓕(X) such that supμ𝓜|∫X f dμ|∊ for every fCb(X, E) with ||f|| ≤ 1 and f|M 0.

  3. supμ∈𝓜 |μ|(X) < ∞ and for every ∊ > 0 there exists M ∈ 𝓕(X) such that supμ∈𝓜 |μ|(X \ M) ≤ .

Proof (i) ⇒ (ii) Assume that (i) holds. Then supμ∈𝓜 |μ|(X) = supμ∈𝓜 ||Φμ|| < ∞ because δβτu. Moreover, there exist υ𝓥δ and a > 0 such that supμ∈𝓜 |∫Xf dμ| ≤ apυ(f) for all fCb(X, E). Hence in view of the proof of Lemma 2.8 there exists a sequence (tn) in X with υ(tn) → 0 and for each μ ∈ 𝓜 there exists (xμ,n)1(E) with n=1xμ,nEa such that

Φμ(f)=Xfdμ=n=1υ(tn)xμ,n(f(tn))forfCb(X,E).

Let > 0 be given. Choose n ∈ ℕ such that υ(tn)εa for nn. Hence for each μ𝓜, n=nε+1υ(tn)xμ,nEε. Let fCb(X, E) with ||f|| ≤ 1 and f|M 0, where M = {t1, , tn). Then for every μ ∈ 𝓜,

|Xfdμ|=|n=nε+1υ(tn)xμ,n(f(tn))|n=nε+1υ(tn)xμ,nEε.

(ii) ⇒ (iii) Assume that (ii) holds. By Lemma 2.5 for μ𝓜 and M ∈ 𝓕(X),

|μ|(X\M)=sup{|Xfdμ|:fCb(X,E)withf1,supp  fX\M}.

It follows that (iii) holds.

(iii) ⇒ (i) Assume that (iii) holds. Then there exists a sequence (Mn) in 𝓕(X) such that MnMn+1 and sup{|μ|(X \ Mn) : μ𝓜} ≤ 2–2n. Let υ(t):=n=𝟙2n1Mn(t) for tX. Then υ𝓥δ. Assume that fCb(X, E) and pυ(f) = suptX υ(t)||f(t)||E ≤ 1. Then sup{||f(t)||E : tMn+1 \ Mn} ≤ 2n, sup{||f(t)E : tM1} ≤ 1 and for every μ ∈ 𝓜, we get

|Φμ(f)|Xf˜d|μ|=M1f˜d|μ|+n=1Mn+1\Mnf˜d|μ||μ|(X)+1.

Let supμ∈𝓜 |μ|(X) = a. Then for fCb(X, E) and every μ ∈ 𝓜, we have |Φμ (f)| ≤ (a + 1)pυ (f), and it follows that (i) holds. □

3 A Riesz representation theory of operators onCb(X, E)

Let Cb(X,E)β denote the bidual of (Cb(X, E), β). Since β-bounded subsets of Cb(X, E) are τu-bounded, the strong topology β(Cb(X,E)β,Cb(X,E)) in Cb(X,E)β coincides with the norm topology in Cb(X, E)′ restricted to Cb(X,E)β. Hence Cb(X,E)β=(Cb(X,E)β,).

Assume that T : Cb(X, E) → F is a (β, || · ||F)-continuous operator. Let T:FCb(X,E)β and T:Cb(X,E)βF stand for the conjugate and the biconjugate operators of T, respectively, i.e., T′(y′) := y′ ◦ T for y′ ∈ F', and T"(Ψ)(y′) := Ψ(y′ ◦ T) for ΨCb(X,E)β(=(Cb(X,E)β,)) and y′ ∈ F′.

Then one can embed B(𝓑o, E) into Cb(X,E)β by the mapping π:B(𝓑o,E)Cb(X,E)β, where for gB(𝓑o, E),

π(g)(Φμ)=(I)XgdμforμM(X,E),

and (I) X g dμ denotes the so-called immediate integral (see [1, 2]). Then

|π(g)(Φμ)|=|(I)Xgdμ|g|μ|(X)=gΦμ

and hence π is bounded and ||π(g)|| ≤ ||g||. One can easily show that

T(y)TyFandT(Ψ)FTΨ,

where Ψ=sup{|Ψ(Φ)|:ΦCb(X,E)β,Φ1}. Then T and T″ are bounded operators, and we define a bounded operator by:

T^:=Tπ:B(𝓑o,E)F.

Define a measure m : 𝓑o → 𝓛(E, F") (called the representing measure of T) by

m(A)(x):=T^(𝟙Ax)forA𝓑o,xE.

For each y′F′, let

my(A)(x):=(m(A)(x))(y)forA𝓑o,xE.

Then my : 𝓑oE′ and by [1, § 4, Proposition 5] for A ∈ 𝓑o we have,

m˜(A)=sup{|my|(A):yBF}.

Then T^ admits an integral representation by the so-called immediate integral (I) X g dm, developed by Foias and Singer [18] and Dinculeanu [1, § 6], [2, § 1], that is,

T^(g)=(I)Xgdm:=limn(I)XsndmforgB(𝓑o,E),

where (sn) is a sequence in 𝓢(𝓑o, E) such that ||sn – g || → 0. Then for y′F′,

T^(g)(y)=((I)Xgdm)(y)=(I)XgdmyforgB(𝓑o,E).

Let iF : FF″ denote the canonical embedding, i.e., iF(y)(y′) = y′(y) for yF, y′F′. Moreover, let jF : iF(F) → F stand for the left inverse of iF, that is, jF ◦ iF = idF.

From the general properties of the operator T^ for hCrc(X, E) we have,

T^(Crc(X,E))iF(F)andT(h)=jF(T^(h))=jF((I)Xhdm).(5)

Hence for each y′F′, we get

y(T(h))=T^(h)(y)=(I)XhdmyforhCrc(X,E).(6)

By M(X, 𝓛(E, F″)) we denote the space of all measures m : 𝓑o → 𝓛(E, F″) such that m˜(X) < ∞ and for each y′F′, myM(X, E′).

Definition 3.1: We say that mM(X, 𝓛(E, F″)) has the tight semivariation if for every ∊ > 0 there exists K𝓚(X) such that m˜(X \ K) ≤ .

Now we can state the following Riesz representation theorem for operators on Cb(X, E).

Theorem 3.2: Assume that T : Cb(X, E) → F is a (β, || · ||F)-continuous operator and m is its representing measure. Then the following statements hold:

  1. mM(X, 𝓛(E, F″)).

  2. For each y′ ∈ F′, y′(T(f)) = ∫X f dmy′ for fCb(X, E).

  3. The mapping F'y′ ↦ myM(X, E′) is (σ(F', F), σ(M(X, E′), Cb(X, E)))-continuous.

  4. m has the tight semivariation.

  5. For every ∊ > 0 there exists K𝓚(X) such that ||T(f)||F∊ if fCb(X, E) with ||f|| ≤ 1 and f|K 0.

  6. m˜(X) = ||T||.

  7. Every fCb(X, E) is m-integrable over X in the Riemann-Stieltjes sense and for y′F′, (Xfdm)(y)=Xfdmy.

  8. X h d m = (I)X h d m for hCrc(X, E).

  9. For fCb(X, E), X f d miF(F) and T(f) = jF(X f d m).

Conversely, assume that a measure mM(X, 𝓛(E, F″)) satisfies the conditions (iii) and (iv). Then there exists a unique (β, || · ||F)-continuous operator T : Cb(X, E) → F such that (ii), (v), (vi), (vii), (viii) and (ix) hold and m coincides with the representing measure of T.

Proof. For y′F′, we have (yT)(f)=XfdμyT for fCb(X, E), where μy◦TM(X, E) (see Theorem 2.4). Then for A𝓑o and xE we have

my(A)(x)=(m(A)(x))(y)=T^(𝟙Ax)(y)=π(𝟙Ax)(yT)=π(𝟙Ax)(ΦμyT)=(I)X(𝟙Ax)dμyT=μyT(A)(x),

i.e., my′ = μy′◦TM(X, E′). Since m˜(X)=T^< (see [1, §9, Theorem 1]), the condition (i) holds. Moreover, y′(T(f)) = ∫X f dmy′ for fCb(X, E), i.e., (ii) holds. Using (i) and (ii) we easily obtain that (iii) holds. Since the family {y′ ◦ T : y′BF′} is β-equicontinuous, by Corollary 2.7 and (ii) for every > 0 there exists K𝓚(X) such that m˜(X\K)=supyBF|my|(X\K)ε and this means that (iv) holds. Then by (iv), (ii) and Lemma 2.5 we easily obtain that (v) holds. Using (ii) and Lemma 2.5, we have

T=sup{|Xfdmy|:fCb(X,E),f𝟙,yBF}=sup{|my|(X):yBF}=m˜(X),

i.e., (vi) holds.

Now we shall show that every f ∈ Cb(X, E) is m-integrable over X. Indeed, let fCb(X, E) and η > 0 be given, and ε=η4m˜(X). By (iv) there exists K𝓚(X) such that supyBF|my|(X\K)η8f.

For each tX let Wt = {sX : ||f(s) – f(t)||E < ∊}. Hence there exists a set {t1, …, tn} in X such that Kj=1n  Wti. Then W:=i=1n  Wti𝓑o and supyBF|my|(X\W)η8f. Define A1 = Wt1, Ai=Wti\j=1i1  Wtj for i = 2, …, n and An+1 = X \ W. Then W=i=1nAi and tiAi for i = 1,..., n and let tn+1An+1. Hence 𝓐 = {A1,..., An, An+1;t1,…tn, tn+1} ∈ 𝓓X. Let 𝓐1, 𝓐2 ∈ 𝓓X and 𝓐1 ≥ 𝓐, 𝓐2 ≥ 𝓐, where, 𝓐1 = {B1,..., Bp,s1,...,sp} and 𝓐2 = {C1,..., Cq, τ1,..., τq}. Let Ji = {j : BjAi} for i = 1, …, n+1. Then arguing similarly as in the proof of Theorem 2.3 one can show that S𝓐ε(f)S𝓐i(f)Fη2 for i = 1, 2, so S𝓐1(f)S𝓐2(f)Fη. This means that (S𝓐(f)) is a Cauchy net in (F″, || · ||F″) and hence the integral X f dm := lim𝓐 S𝓐(f) exists in F″. This means that f is m-integrable over X.

Note that for each y′ ∈ F' and 𝓐 = {A1,...,An;t1,...,tn) ∈ 𝓓X we have S𝓐(f)(y)=i=1nmy(Ai)(f(ti)) and X f dmy = lim𝓐 S𝓐(f)(y′). Hence we get,

(Xfdm)(y)=lim𝓐S𝓐(f)(y)=Xfdmy.(7)

Thus (vii) holds. Now we shall show that the Integration operator Sm : Cb(X, E) → F″ defined by the equation

Sm(f):=XfdmforfCb(X,E)

is (β, || · ||F)-continuous. Indeed, let > 0 be given. Since {y′ ◦ T : y′ ∈ BF} is β-equicontinuous and (ii) holds, there exists a neighbourhood V of 0 for β such that supyBF|X  f  dmy|ε for fV. Assume that (fα) is a net in Cb(X, E) such that fα → 0 for β. Then one can choose α such that fαV for αα. Hence for αα we have,

Sm(fα)F=sup{|(Xfαdm)(y)|:yBF}=sup{|Xfαdmy|:yBF}ε,

i.e., Sm is (β, || · ||F)-continuous. Using (ii), (7) and (6) for hCrc(X, E) and y′F, we have

Sm(h)(y)=(Xhdm)(y)=Xhdmy=y(T(h))=T^(h)(y).

It follows that Sm(h)=T^(h), where T^(h)iF(F). Hence

Xhdm=(I)XhdmiF(F).

Thus the condition (viii) is satisfied. Assume now that f ∈ Cb(X, E) and choose a net (hα) in Crc(X, E) such that hαf for β. Then

Xfdm=Sm(f)=limαSm(hα)=limαXhαdm,

and hence X f dmiF(F) because X hαdmiF(F). It follows that

T(f)=limαT(hα)=limαjF(Xhαdm)=jF(limαXhαdm)=jF(Xfdm),

i.e., (ix) holds.

Assume that mM(X, 𝓛(E, F)) satisfies the conditions (iii) and (iv). For fCb(X, E) define a linear mapping Ψf on F by Ψf(y′) = ∫X f dmy for y′F. Then Ψf is σ(F, F)-continuous, so there exists a unique yfF such that Ψf = iF(yf),i.e., Ψf(y′) = y(yf) for y′F. Define a mapping T : Cb(X, E) → F by T(f) := yf. Then for y′ ∈ F′, (y′ ◦ T)(f) = y′(yf) = ∫X f dmy for fCb(X, E), i.e., (ii) holds. Hence by Corollary 2.7 the set {y′ ◦ T : y′ ∈ BF} is β-equicontinuous, and this means that T is (β, || · ||F)-continuous.

Let mo : 𝓑o𝓛(E, F") be the representing measure of T. Then by the first part of theorem moM(X, 𝓛(E, F")) and for each y′ ∈ F′, we have y′(T(f)) = ∫X f d(mo)y for fCb(X, E). It follows that (mo)y = my′, so mo = m. In view of the first part of the theorem, the conditions (v), (vi), (vii), (viii) and (ix) are satisfied. □

Assume that T : Cb(X, E) → F is a (β, || ·||F)-continuous operator and m is its representing measure. For xE let us set,

Tx(u):=T(ux)foruCb(X)andmx(A):=m(A)(x)forA𝓑o.

Then Tx : Cb(X) → F is a (β, || · ||F)-continuous operator. Let χ:B(𝓑o)Cb(X)β stand for the canonical embedding, i.e., for υB(𝓑o),

χ(υ)(φv):=(I)XυdvforvM(X),

where φv(u) = Xu dv for uCb(X). Define

T^x:=(Tx)χ:B(𝓑o)F.

Then T^x(Cb(X))iF(F) and Tx(u)=jF(T^x(u)) for uCb(X).

Lemma 3.3: Let T : Cb(X, E) → F be a (β, || · ||F)-continuous operator and m be its representing measure. Then for each xE the following statements hold:

  1. T"(π(𝟙Ax)) = (Tx)"(χ(𝟙A)) for A𝓑o.

  2. mx(A)=T^x(1A) for A𝓑o.

  3. T^x(υ)=(I)X  υdmx for υB(𝓑o).

Proof. Let y′F. Then by Theorem 3.2 for uCb(X), we have

(yTx)(u)=y(T(ux))=X(ux)dmy=Xudmx,y=φmx,y(u),

where φmx,yCb(X)β. Hence, we get

(Tx)(χ(𝟙A))(y)=χ(𝟙A)((Tx)(y))=χ(𝟙A)(yTx)=χ(𝟙A)(φmx,y)=mx,y(A)=mx(A)(y).

On the other hand, by Theorem 3.2, y    T=Φmy, and hence

T(π(𝟙Ax))(y)=π(𝟙Ax)(T(y))=π(𝟙Ax)(yT)=π(𝟙Ax)(Φmy)=(I)X(𝟙Ax)dmy=mx(A)(y).

It follows that (i) holds. For A𝓑o, we have

mx(A):=T^(𝟙Ax)=T(π(𝟙Ax))=(Tx)(χ(𝟙A))=T^x(𝟙A),

and hence, T^x(υ)=(I)  X  υ  dmx for υB(𝓑o). Thus (ii) and (iii) are satisfied. □

Dobrakov [20, Theorem 2] showed that if X is a locally compact Hausdorff space and T : C0(X, E) → F is a bounded linear operator, then its representing measure takes its values in 𝓛(E, F) if and only if Tx : C0(X) → F is weakly compact for each xE. Now we prove the analogue of this result for (β, || · ||F)-continuous operators T : Cb(X, E) → F, where X is a completely regular Hausdorff space (see [26, Theorem 4.4]).

Theorem 3.4: Let T : Cb(X, E) → F be a (β, || · ||F)-continuous operator and m be its representing measure. Then for each xE the following statements are equivalent:

  1. Tx : Cb(X) → F is weakly compact.

  2. m(A)(x) ∈ iF(F) for each A𝓑o.

  3. mx : 𝓑oF″ : countably additive.

  4. mx : 𝓑o → Fis strongly bounded.

Proof. (i) ⇒ (ii) Assume that Tx is weakly compact. in view of [42, Corollary 9.3.2] it follows that (Tx)(Cb(X)β)iF(F). Hence T^x(B(𝓑o))iF(F) and by Lemma 3.3 mx(A) ∈ iF(F) for A ∈ 𝓑o.

(ii) ⇒ (iii) Assume that m(A)(x) ∈ iF(F) for A𝓑o, and let (mF)x(A) := jF(m(A)(x)) for A𝓑o. Then (y′ ◦ (mF)x)(A) = (m(A)(x))(y) = mx, y (A) for A𝓑o. in view of Theorem 3.2 for each y′ ∈ F′, my′M(X, E′) and it follows that mx,y is countably additive. Hence y (mF)x is countably additive, and by the Orlicz-Pettis theorem (mF)x : 𝓑oF is countably additive. Hence mx : 𝓑oF″ is also countably additive.

(iii) ⇒ (iv) It is obvious.

(iv) ⇒ (i) Assume that mx : 𝓑oF″ is strongly bounded. Then in view of Lemma 3.3 and [45, Theorem 1,

p. 148] T^x : B(𝓑o) → F″ is weakly compact. Since Tx(u) = jF(T^x(u)) for uCb(X), we obtain that Tx is weakly compact. □

Assume that T : Cb(X, E) → F is a (β, || · ||F)-continuous operator such that Tx : Cb(X) → F is weakly compact for each xE. From now on we will write:

mF(A)(x):=jF(m(A)(x))forA𝓑o,xE.

Then mF : 𝓑o → 𝓛(E, F) and mF˜(A)=m˜(A) for A ∈ 𝓑o.

By 𝓣s we denote the topology of simple convergence in ca(𝓑o). Then 𝓣s is generated by the family {PA : A𝓑o} of seminorms, where PA(v) = |v(A)| for vca(𝓑o). Note that M(X) is a closed subspace of the Banach space ca(𝓑o) (see Remark 2.6).

A completely regular Hausdorff space X is said to be a k-space if each set which meets every compact subset in a closed set must be closed. X is a k-space, for instance if X is locally compact or first countable (see [38, Chap. 3, § 3], [6, p. 107]). Making use of [43] we can state the following extension to k-spaces of the celebrated Dieudonné-Grothendieck’s criterion on relative weak compactness in the space M(X) (see [44, Theorem 2], [45, Theorem 14, p. 98-103]), which will play a crucial role in the study of operators on Cb(X, E).

Theorem 3.5: Assume that X is a k-space and 𝓜 is a subset of M(X) such that supv∈𝓜 |v|(X) < ∞. Then the following statements are equivalent:

  1. 𝓜 is relatively weakly compact in the Banach space M(X).

  2. 𝓜 is uniformly countably additive, i.e., supv𝓜 |v(An)| → 0 whenever An ∅, (An) ⊂ 𝓑o.

  3. 𝓜 is uniformly strongly additive, i.e., supv𝓜 |v(An)| → 0 whenever (An) is pairwise disjoint in 𝓑o.

  4. supv𝓜 |v|(An) → 0 whenever (An) is a pairwise disjoint sequence in 𝓑o.

  5. 𝓜 is a relatively 𝓣s-compact subset of M(X).

  6. 𝓜 is uniformly regular, i.e., for every A𝓑o and ∊ > 0 there exist K𝓚(X) and O ∈ 𝓣 with KAO such that supv∈𝓜 |v|(O \ K) ≤ .

  7. supv∈𝓜 |v|(On) → 0 whenever (On) is a pairwise disjoint sequence in 𝓣.

Proof. (i) ⇔ (ii) It follows from [46, Chap. 7, Theorem 13] because M(X) is a closed subset of the Banach space ca(𝓑o) (see [47, Chap. 3, §3, Corollary 3]).

(ii) ⇔(iii) See [46, Theorem 10, pp. 88-89].

(iii) ⇔(iv) See [45, Proposition 17, p. 8].

(iv) ⇔ (v) ⇔ (vi) It follows [43, Remark, pp. 211-212]. (iv) ⇔ (vii) It is obvious.

(vii) ⇔ (iv) Assume that (vii) holds. Then by [43, Theorem 8] the set {|v| : v𝓜} is relatively 𝓣s-compact In M(X). Hence by [43, Remark, pp. 211-212] the condition (iv) holds. □

4 Strongly bounded operators onCb(X, E)

Let T : Cb(X, E) → F be a (β, || · ||F)-continuous operator and m be its representing measure. Then T is said to be strongly bounded if m˜ is strongly bounded, i.e., m˜(An) → 0 whenever (An) is a pairwise disjoint sequence in 𝓑o. Note that m˜ is strongly bounded if and only if {|my| : yBF′} is uniformly countably additive, i.e., sup{|my|(An) : yBF′} → 0 whenever An ↓ ∅, (An) ⊂ 𝓑o (see [46, Theorem 10, pp. 88-89]), i.e., m˜ is continuous at ∅.

For x ∈ E we have, ||mx(A)||Fm˜(A)||x||E for A ∈ 𝓑o. It follows that if T : Cb(X, E) → F is a (β || · ||F)-continuous strongly bounded operator, then for each xE, mx : 𝓑oF″ is strongly bounded, and hence by Theorem 3.4, m(A)(x) ∈ iF(F) for A ∈ 𝓑o and Tx : Cb(X) → F is a weakly compact operator. Then we can define a measure mF : 𝓑o → 𝓛(E, F) by

mF(A)(x):=jF(m(A)(x)).

Now we can state an intrinsic characterization of (β, || · ||F)-continuous strongly bounded operators T : Cb(X, E) → F by the Grothendieck’s condition in [44] (see (ii) below).

Theorem 4.1: Assume that X is a k-space. Let T : Cb(X, E) → F be a (β, || · ||F)-continuous operator and m be its representing measure. Then the following statements are equivalent:

  1. T is strongly bounded, i.e., m˜ is continuous at ∅.

  2. T(fn) → 0 whenever (fn) is a uniformly bounded sequence in Cb(X, E) such that fn(t) → 0 in E for each tX.

  3. T(fn) → 0 whenever (fn) is a uniformly bounded sequence in Cb(X, E) such that supp fn ∩ supp fk = ∅ for nk.

  4. For every A and ∊ > 0 there exist K𝓚(X) and O ∈ 𝓣 such that KAO and m˜(O \ K) ≤ .

  5. T is sequentially.(δ, || · ||F)-continuous.

Proof. (i) ⇒ (ii) Assume that T is strongly bounded. Let λ ∈ ca+(𝓑o) be a control measure for {|my| : y′ ∈ BF} (see [45, Theorem 4, pp. 11-12]). Let (fn) be a sequence in Cb(X, E) such that supn ||fn|| = M < ∞ and fn(t) → 0 for every tX. Then for a given > 0 there exists η > 0 such that sup{|my|(A) : yBF} ≤ ε2M whenever λ(A) ≤ η, A ∈ 𝓑o. Since f˜nB(𝓑o), by the Egoroff theorem there exists Aη ∈ 𝓑o with λ(X \ Aη) ≤ η and suptAηf˜n(t)ε2m˜(X) for nn for some n ∈ ℕ. For each y′ ∈ BF′ and nn by Theorem 3.2 and (1), we have

|y(T(fn))|=|Xfndmy|Xf˜nd|my|=Aηf˜nd|my|+X\Aηf˜nd|my|ε2m˜(X)|my|(Aη)+M|my|(X\Aη)ε2m˜(X)|my|(X)+Mε2Mε2+ε2=ε.

Hence ||T(fn)||F for nn, as desired.

(ii) ⇒ (iii) It is obvious.

(iii) ⇒ (i) Assume that (iii) holds and T is not strongly bounded. Then by Theorem 3.5 there exist 0 > 0, a pairwise disjoint sequence (On) in 𝓣 and a sequence (yn) in BF′ such that |myn|(On)>ε0. Hence by Lemma 2.5 there exists a sequence (hn) in Cb(X) ⊗ E with ||hn|| = 1 and supp hnOn for n ∈ ℕ such that

OnhndmynmynOnε02>ε02.

Then for n ∈ ℕ,

T(hn)F=sup{|y(T(hn))|:yBF}=sup{|Xhnd  my|:yBF}=sup{|Onhn  d  my|:yBF}|Onhn  d  myn|>ε02.

On the other hand, since supp hn ∩ supp hk = ∅ for nk, we get ||T(hn)||F → 0. This contradiction establishes that (i) holds.

(i) ⇔ (iv) It follows from Theorem 3.5 for 𝓜 = {|my| : y′ ∈ BF}.

(ii) ⇔ (v) It is obvious because for a sequence (fn) in Cb(X, E), fn 0 for δ if and only if supn||fn|| < ∞ and fn(t) → 0 for every tX. □

Remark 4.2: Incase X is a compact Hausdorff space, the equivalence (i) ⇔ (ii) of Theorem 4.1 was obtained by Brooks and Lewis (see [27, Theorem 2.1]).The following Riesz representation type theorems will be of importance in the study of operators on Cb(X, E).

Theorem 4.3: Assume that X is a k-space and mM(X, 𝓛(E, F)) has the regular semivariation. Then the following statements hold:

  1. The mapping T : Cb(X, E) → F defined by the equation: T(f) = ∫X f dm is a strongly bounded operator.

  2. T(h) = ∫X h dm = (I) X h dm for hCrc(X, E).

Proof. In view of Theorem 2.3 and Corollary 2.7 the family {y ◦ T : yBF} is β-equicontinuous and this means that T is (β, || · ||F)-continuous. By Theorem 4.1 T is strongly bounded, so (i) is satisfied. In view of Theorems 2.3 and 3.2 for yF, we get

y(T(h))=y(Xh  dm)=y((I)Xh  dm)  for  hCrc(X,E),

and if follows that (ii) holds. □

Corollary 4.4: Assume that X is a k-space. Let T : Cb(X, E) → F be a strongly bounded operator and m be its representing measure. Then the following statements hold

  1. mFM(X, 𝓛(E, F)) and mF has the regular semivariation.

  2. Every fCb(X, E) is mF-integrable over all A𝓑o.

  3. T(f) = ∫X f dmF for fCb(X, E).

  4. T(h) = ∫X h dmF = (I) X h dmF for hCrc(X, E).

Proof. By Theorems 4.1 and 4.3 and 2.3 the statements (i) and (ii) are satisfied. Using Theorems 3.2 and 4.3 for each yF, we have

y(T(f))=Xf  dmy=y(Xf  dmF)  for  fCb(X,E).

and it follows that (iii) holds. The condition (iv) follows from (iii) and Theorem 4.3. □

Definition 4.5: (i) A measure mM(X, 𝓛(E, F)) is said to be discrete if for every yF, my′Md(X, E). By Md(X, 𝓛(E, F)) we will denote the subspace of M(X, 𝓛(E, F)) of all discrete measures.(ii) We say that mMd(X, 𝓛(E, F)) has the discrete semivariation if for every ∊ > 0 there exists M𝓕(X) such that m˜(X \ M) ≤ .

Theorem 4.6: Assume that X is a k-space. Let T : Cb(X, E) → F be a (δ, || · ||F)-continuous linear operator and m be its representing measure. Then the following statements hold:

  1. T is strongly bounded.

  2. mFMd(X, 𝓛(E, F)) and mF has the discrete semivariation.

Proof. Since δβ, by Theorem 4.1 (i) holds. Then by Corollary 4.4, mFM(X, 𝓛(E, F)) and by Theorem 2.10 my′Md(X, E) for each y′ ∈ F. Hence mFMd(X, 𝓛(E, F)). Moreover, since the family {y′ ◦ T : yBF′} is δ-equicontinuous, by Corollary 2.11 the family {|my′| : yBF′} is uniformly discrete, that is, mF has the discrete semivariation, i.e., (ii) holds. □

Now using Theorem 3.5 and arguing as in the proof of Corollary 13 in [36] we can show a related result for (β, || · ||F)-continuous operators T : Cb(X, E) → F (see also [45, Theorem 15, pp. 159-160]).

Corollary 4.7: Assume that X is a k-space and F contains no isomorphic copy of co. Then every (β, || · ||F)-continuous operator T : Cb(X, E) → F is strongly bounded.

5 Radon extension of operators onCb(X, E)

For λ ∈ M+(X) let 𝓛(λ, E) stand for the vector-space of all λ-measurable functions g : XE such that ||g|| := ess suptX||g(t)||E < ∞.

Proposition 5.1: Assume that λM+(X). Then Cb(X, E) ⊂ 𝓛(λ, E).

Proof. By the regularity of λ there exists a sequence (Kn) in 𝓚(X) such that Kn ↑ and λ(X\Kn)<1n for n ∈ ℕ. Then for A=n=1  Kn, we have λ(X \ A) = 0. Assume that fCb(X, E) and let hn = 𝟙Knf for n ∈ ℕ. Note that by the Pettis measurability theorem, hn is λ-measurable (see [45, Theorem 2, p. 42]) and hn(t) → f(t) for tA. It follows that f is λ-measurable and f ∈ 𝓛(λ, E). □

Assume that mM(X, 𝓛(E, F)) has the regular semivariation and let λM+(X) be a control measure for {|my′| : yBF′}. We can assume that λ to be complete (if necessary we can take the completion (X, 𝓑o, λ) of (X, 𝓑o, λ), and for each yBF′, we extend |my′| to 𝓑o). It is known that if g ∈ 𝓛(λ, E), then one can define the Radon-type integral of g with respect to m by the equation:

(R)Xg  dm:=limn(R)Xsn  dm,

where (sn) is a sequence in 𝓢(𝓑o, E) such that ||sn(t) – g(t)||E → 0 λ-a.e. on X and ||sn(t)||E ≤ ||g(t)||E λ-a.e. on X (see [48, § 1]). Then the Radon integration operator Tm : 𝓛(λ, E)—?F defined by the equation:

Tm(g):=(R)Xg  dm

is λ-σ-smooth, i.e., Tm(gn) → 0 whenever (gn) is a sequence in 𝓛(λ, E) such that supn ||gn|| < ∞ and ||gn(t)||E → 0 λ-a.e. on X (see [48, Proposition 3.5]). Note that for y′ ∈ F and g𝓛(λ, E), we have

y(Tm(g))=(R)Xg  dmy  and  |(R)Xg  dmy|Xg˜  d|my|.(8)

Now we can state a Radon extension theorem for operators on Cb(X, E), which will be useful in the study of operators on Cb(X, E) (see the proof of Corollary 7.11 below).

Theorem 5.2: Assume that X is a k-space. Let T : Cb(X, E) → F be a strongly bounded operator and m be its representing measure. If λM+(X) is a control measure for {|my′ : y′BF′ }, then T possesses a λ-σ-smooth Radon extension TmF : 𝓛∞(λ, E) → F such that T(f)=TmF(f)=(R)Xf  dmF=Xf  dmF    for    fCb(X,E),and for each y′ ∈ F′, there exists a weak* -measurable function gy : XEsuch that ||gy(·)||E ∈ 𝓛1(λ) and y(T(f))=Xf,gydλ    for    fCb(X,E)    and    y    T=Xgy(t)Edλ.

Proof. By Corollary 4.4 mFM(X, 𝓛(E, F)) and mF has the regular semivariation. Moreover, every fCb(X, E) is mF-integrable and T(f) = ∫X f dmF.

Assume that y′ ∈ F' and (fα) is a net in Cb(X, E) such that fα → 0 for β. Since Cb(X, E) ⊂ 𝓛(λ, E) (see Proposition 5.1), using (8), we have

|y(TmF(fα))|=|(R)Xfα  dmy|Xf˜α  d|my|.

Note that f˜α0 for β in Cb(X) and |my′| M(X). Hence Xf˜αd|my|0. It follows that y    (TmF|Cb(X,E)) is a β-continuous functional. For hCrc(X, E) ⊂ B(𝓑o, E), we get

y(TmF(h))=(R)Xh  dmy=(I)Xh  dmy=y(T(h)).

Since Crc(X, E) is β-dense in Cb(X, E), for each y′ ∈ F, we have y(TmF(f)) = y(T(f)) for every fCb(X, E). Hence

T(f)=TmF(f)=(R)Xf  dmF=Xf  dmF    for    fCb(X,E).

Let yF. Then according to the Radon-Nikodym type theorem (see [49, Theorem 1.5.3]) there exists a weak*-measurable function gy : XE such that ||gy(·)||E ∈ 𝓛1(λ) and for every xE and A𝓑o,

my(A)(x)=Ax,gy    and    |my|(A)=AgyEdλ.(9)

Now let fCb(X, E) ⊂ 𝓛 (λ, E). Then there exists a sequence (sn) in 𝓢(𝓑o, E) such that ||sn(t) – f(t)||E → 0 λ-a.e. and ||sn(t)||E||f(t)||E λ-a.e. (see [2, Theorem 1.6, p. 4]). One can easily show that

limnXsn,gydλ=Xf,gydλ.

It follows that

y(T(f))=Xf  dmy=(R)Xf  dmy=limn(R)Xsn  dmy=limnXsn,gydλ=Xf,gydλ.

Note that by (4), we get ||y′ ◦ T|| = |my′|(X) = ∫X ||gy′(t)||E′ . □

Remark 5.3: If E is a separable Banach space and Cb(X) ⊗ E is supposed to be βσ-dense in Cb(X, E), then a related result to Theorem 5.2 for (βσ, || · ||F)-continuous operators T : Cb(X, E) → F can be found in [37, Proposition 16].

6 Weakly compact operators onCb(X, E)

If X is a compact Hausdorff space (resp. X is a locally compact Hausdorff space), weakly compact operators T : Cb(X, E) → F (resp. T : Co(X, E) → F) have been studied intensively by Batt and Berg [21, 23], Brooks and Lewis [26], Bombal and Cembranos [24] and Saab [29].

Recall that a linear operator T : Cb(X, E) → F is called (β, || · ||F)-weakly compact if there exists a neighbourhood V of 0 for β such that T(V) is relatively weakly compact in F. One can easily show that a (β, || · ||F)-weakly compact operator T : Cb(X, E) → F is (β, || · ||F)-continuous.

The following characterization of weakly compact operators on (Cb(X, E), β) will be useful.

Theorem 6.1: Fora (β, || · ||F)-continuous linear operator T : Cb(X, E) → F the following statements are equivalent:

  1. T is (β, || · ||F)-weakly compact.

  2. T maps τu-bounded sets in Cb(X, E) onto relatively weakly compact sets in F.

  3. T(Cb(X,E)β)iF(F).

  4. T′(BF′) is relatively σ(Cb(X,E)β,Cb(X,E)β)-compact in Cb(X,E)β.

Proof. (i) ⇒ (ii) It is obvious because βτu.

(ii) ⇒ (i) Assume that (ii) holds. Since β-bounded subsets of Cb(X, E) are τu-bounded, T transforms β-bounded sets into relatively weakly compact sets in F. But .Cb(X, E), β) is a generalized DF-space, so by [50, Theorem 3.1] T is (β, || · ||F)-weakly compact, as desired.

(ii) ⇔ (iii) ⇔ (iv) It follows from [42, Theorem 9.3.2]. □

Now we extend a characterization of weakly compact operators T : Co(X, E) → F of [26, Theorem 4.1] to (β, || · ||F)-continuous weakly compact operators T : Cb(X, E) → F.

Theorem 6.2: Let T : Cb(X, E) → F be a (β, || · ||F)-continuous operator and m be its representing measure. Then the following statements hold:

  1. Assume that T is (β, || · ||F)-weakly compact. Then T is strongly bounded and for each A𝓑o, mF(A) : EF is a weakly compact operator.

  2. Assume that E and E″ have the RNP and T is strongly bounded and for each A𝓑o, mF(A) : EF is a weakly compact operator. Then T is (β, || · ||F)-weakly compact.

Proof. (i) In view of Theorem 6.1 the conjugate operator T:FCb(X,E)β maps BF′ onto a relatively weakly compact set in the Banach space (Cb(X,E)β,), where T(y)(f) = ∫X f dmy′ for yF, fCb(X, E). Then {my′ : yBF′} is a relatively weakly compact subset of the Banach space M(X, E). Hence {my′ : yBF′} is a relatively weakly compact subset of the Banach space (cabv(𝓑o; E′), || · ||), where ||μ|| = |μ|(X). According to the Bartle-Dunford-Schwartz theorem (see [45, Theorem 5, pp. 105-106]), we obtain that the set {|my| : yBF′} is uniformly countably additive and for each A𝓑o, the set {my′(A) : yBF′} is relatively weakly compact in E. It follows that T is strongly bounded and since for yF, mF(A)(y) = my′(A), we derive that mF(A) : FE is weakly compact, and hence mF(A) : EF is weakly compact.

(ii) Since T is strongly bounded, {|my′| : yBF′} is uniformly countably additive. Moreover, for each A𝓑o, {my′(A) : yBF′} is relatively weakly compact in E. This means that {my′ : yBF′} is a relatively weakly compact set in the Banach space cabv(𝓑o, E′) (see [45, Theorem 5, pp. 105-106]). But {my′ : yBF′M(X, E) and M(X, E) is closed in the Banach space cabv(𝓑o, E). Hence {my′ : yBF′} is relatively weakly compact subset of (M(X, E′), || · ||) (see [47, Chap. 3, §3, Corollary 3]). It follows that {Φmy:yBF} is a relatively weakly compact subset of the Banach space {my:yBF} According to Theorem 6.1 T is (β, || · ||F)-weakly compact. □

Remark 6.3: In[37, Theorem 29] we obtain a related result to Theorem 6.2 for (βσ, || · ||F)-continuous operators T : Cb(X, E) → F, where Cb(X) ⊗ E is supposed to be βσ -dense in Cb(X, E).

7 Unconditionally converging operators on Cb(X, E)

We start with the following definition.

Definition 7.1: Assume that (Z, ξ) is a locally convex Hausdorff space and (F, || · ||F) is a Banach space. A (ξ, || · ||F)-continuous linear operator T : ZF is said to be unconditionally converging if the series n=1T(zn) converges unconditionally in F whenever n=1|z(zn)|< for every zZξ.

By 𝓕(ℕ) we denote the family of all finite subsets of ℕ.

Lemma 7.2: For a sequence (fn) in Cb(X, E) the following statements are equivalent:

  1. sup{||∑iMfi|| :M𝓕(ℕ)} < ∞.

  2. n=1|Φ(fn)|< for all Φ ∈ Cb(X, E).

  3. n=1|Xfn  dμ|< for all μM(X, E).

Proof. (i) ⇔ (ii) It is well known (see [46, Chap. 5, Theorem 6, p. 44]).

(ii) ⇒ (iii) It is obvious because βτu.

(iii) ⇒ (i) Assume that (iii) holds. Then for M𝓕(ℕ) and μM(X, E), we have

|X(iMfi)dμ|=|iMXfi  dμ|iM|Xfi  dμ|±n=1|Xfn  dμ|<.

This means that the set {i∈M fi:M ∈ 𝓕(ℕ)} is σ(Cb(X,E),Cb(X,E)β)-bounded, and hence it is β-bounded. If follows that sup{||∑iMfi|| : M ∈ 𝓕(ℕ)} < ∞. □

Proposition 7.3: Let T : Cb(X, E) → F be a (β, || · ||F)-continuous linear operator. Then the following statements are equivalent:

  1. T is unconditionally converging.

  2. T(fn) → 0 whenever n=1|Xfn  dμ|< for every μM(X, E′).

Proof. (i) ⇒ (ii) It is obvious.

(ii) ⇒ (i) Assume that (ii) holds and T is not unconditionally converging. Then in view of Lemma 7.2, by [46, Exercise 8(i), p. 54] there exists a subspace S of Cb(X, E) isomorphic to co such that T|S : SF is an isomorphism. Let I : coCb(X, E) be an embedding and fn = I(en) ∈ S for n ∈ ℕ, where {en : n ∈ ℕ} is the canonical base in co. Then for μM(X, E′), Φμ ◦ I ∈ (co)′, so there exists a unique (si) ∈ 1, such that (Φμ    I)((ri))=i=1risi for every (ri) ∈ co. Hence we get n=1|sn|<.. On the other hand, there exist a > 0 and b > 0 such that ||T(fn)||Fa||fn|| ≥ ab ||en|| = ab. This contradiction established that (i) holds. □

If X is a compact Hausdorff space, Swartz [51] proved that every unconditionally converging operator T : C(X, E) → F is strongly bounded. Dobrakov (see [20, Theorem 3]) showed that if X is a locally compact Hausdorff space, then every unconditionally converging operator T : C0(X, E) → F is strongly bounded and for every Borel set A in X, the operator m(A) : EF is unconditionally converging. We extend this result to the setting when T : Cb(X, E) → F is a (β, || · ||F)-continuous operator and X is a k-space.

Theorem 7.4: Assume that X is a k-space. Let T : Cb(X, E) → F be a unconditionally converging operator and m stand for its representing measure. Then the following statements hold:

  1. T is strongly bounded.

  2. For every A𝓑o, mF(A) : EF is unconditionally converging.

Proof. (i) Assume that (fn) is a sequence in Cb(X, E) such that supn ||fn|| = a < ∞ and supp fn ∩ supp fk = ∅ for nk. Then for each μM(X, E), we have

n=1|Xfn  dμ|n=1|Xf˜n  d|μ||an=1|μ|(supp  fn)=a|μ|(n=1supp  fn)a|μ|(X).

By Proposition 7.3 T(fn) → 0; and hence by Theorem 4.1 T is strongly bounded,

(ii) Let A ∈ 𝓑o and > 0 be given. in view of Theorem 4.1 there exist K ∈ 𝓚(X) and O ∈ 𝓣 with KAO such that m˜(O \ K) ≤ . Then one can choose υCb(X) with 0 ≤ υ ≤ 𝟙X, υ|K 1 and v|X \ O 0. Define Tv(x) := T(vx) for xE. To show that Tv : EF is unconditionally converging, assume that n=1|X  fn  dμ|=n=1|(Φμ    I)(en)|=n=1xn is a weakly unconditionally Cauchy sequence in a Banach space E. Then for M𝓕(ℕ), || ∑iM(υxi)|| ≤ ∑iMxi||E < ∞ (see [45, p. 150]), and by Lemma 7.2 n=1|X(υxn)dμ|< for all μM(X, E). Hence n=1T(υxn) is unconditionally convergent, i.e., Tv : EF is unconditionally converging. For each xBE by Corollary 4.4, we have

Tυ(x)mF(A)(x)F=(I)X((υ𝟙A)dmF)F=sup{|y(I)X((υ𝟙A)dmF)|:yBF}sup{X|υ𝟙A|d|my|:yBF}sup{|my|(O\K):yBF}=m˜(O\K)ε.

Hence ||TυmF(A)|| and since the set of all unconditionally converging operators from E to F is a closed linear subspace of (𝓛(E, F), || · ||) (see [20, p. 20]), we derive that mF(A) is unconditionally converging. □

Brooks and Lewis [26, Theorem 5.2] showed that if X is a locally compact Hausdorff space and E contains no isomorphic copy of co then every strongly bounded operator T : Co(X, E) → F is unconditionally converging. Now we extend this result to the setting when X is a k-space and T : Cb(X, E) → F is a (β, || · ||F)-continuous operator.

Theorem 7.5: Assume that X is a k-space and E contains no isomorphic copy of co. Let T : Cb(X, E) → F be a strongly bounded operator. Then T is unconditionally converging.

Proof. Let m be a representing measure of T. By Corollary 4.4 mFM(X, 𝓛(E′, F)) has the regular semivariation and let λM+(X) be a control measure for {|my′| : yBF}. Then the Radon integration operator TmF : 𝓛(λ, E) → F is λ-σ-smooth (see Theorem 5.2). Assume that (fi) is a sequence in Cb(X, E) such that i=1|Xfi  dμ|< for each μM(X, E′). Hence i=1|x(fi(t))|< for each tX, xE because δt,xCb(X,E)β. It follows that i=1fi(t) is an unconditionally convergent series in E for each tX because E contains no isomorphic copy of co (see [52]). Hence fn(t) → 0 for tX and sup || fn|| < ∞, and by Theorem 5.2 T(fn) = (R) X fn dmF → 0. In view of Proposition 7.3 it means that T is unconditionally converging. □

Remark 7.6: The related results to Theorems 7.4 and 7.5 for (βσ, || · ||F)-continuous operators T : Cb(X, E) → F can be found in [37, Theorems 17 and 18].

As a consequence of Theorems 4.1, 7.4 and 7.5 we have the following result.

Corollary 7.7: Assume that X is a k-space and E is a reflexive Banach space. Then for a (β, || · ||F)-continuous linear operator T : Cb(X, E) → F the following statements are equivalent:

  1. T is sequentially (δ, || · ||F)-continuous.

  2. T is (β, || · ||F)-weakly compact.

  3. T is unconditionally converging

Following [28, 53] we have a definition.

Definition 7.8: A locally convex Hausdorff space (Z, ξ) is said to have the V property of Pełczynski if for every Banach space (F, || · ||F), every unconditionally converging operator T : ZF is (ξ, || · ||F)-weakly compact.

As a consequence of Corollary 7.7, we have:

Corollary 7.9: Assume that X is a k-space and E is reflexive. Then the space (Cb(X, E), β) has the V property of Pełczynski.

Remark 7.10: If X is a compact Hausdorff space, the related results to Corollary 7.9 were obtained by Cembranos, Kalton, E. Saab and P. Saab [53, Theorem 3] (see also [32, Corollary 2.7], [30, Corollary 4]).

As an application of Theorem 5.2, we get the following characterization of unconditionally converging operators on Cb(X, E).

Corollary 7.11: Assume that X is a k-space. Let T : Cb(X, E) → F be a strongly bounded operator and m be its representing measure. If λM+(X) is a control measure for {|my| : yBF′}, then the following statements are equivalent:

  1. T is unconditionally converging.

  2. limnXfn,gyndλ=0 whenever n=1|X  fn  dμ|< for every μM(X, E) and ynBF for n ∈ ℕ.

Proof. (i) ⇒ (ii) Assume that (i) holds and let (fn) be a sequence in Cb(X, E) such that n=1|X  fn  dμ|< for every μM(X, E) and (yn) is a sequence in BF′. Then by Theorem 5.2, we have

|Xfn,gyndλ|=|yn(T(fn))|T(fn)F0.

(ii) ⇒ (i) Assume that (ii) holds. Let (fn) be a sequence in Cb(X, E) such that n=1|X  fn  dμ|< for every μM(X, E′). Choose a sequence (yn) in BF′ such that |yn(T(fn))|12T(fn))F. Then by Theorem 5.2, we have

T(fn)F2|yn(T(fn))|=2|Xfn,gyndλ|0.

Using Proposition 7.3 we obtain that T is unconditionally converging. □

By Md(X) we denote the subspace of M(X) of all discrete measures. Note that if XMd(X), then λ=n=1cnνtn, where tnX and n=1|cn|< and ∫X udvtn = u(tn) for uCb(X). Then vtn(A) = 𝟙A(tn) for A𝓑o.

Theorem 7.12: Assume that X is a k-space. Let T : Cb(X, E) → F be a strongly bounded operator and m be its representing measure. If λMd+(X) is a control measure for {|my′| : yBF′}, then the following statements are equivalent:

  1. T is unconditionally converging.

  2. For every A𝓑o, mF(A) : EF is : unconditionally converging.

Proof. (i) ⇒ (ii) See Theorem 7.4.

(ii) ⇒ (i) Assume that (ii) holds and λ=n=1cnνtn with cn ≥ 0 and n=1cn<. Let (xn) be a sequence in E such that n=1|x(xn)|< for every xE and (yn) be a sequence in BF′. In view of (9) for every A𝓑o, we have

|Axn,gyndλ|=|myn(A)(xn)|mF(A)(xn)F,

where ||mF(A)(xn)||F → 0. It follows that for each i ∈ ℕ, limnxn,gyn(ti)=limn  {ti}xn,gyndλ=0. Assume now that (fn) is a sequence in Cb(X, E) such that n=1|X  fn  dμ|< for every μM(X, E′). Then for x′ ∈ E′, we have i=1|x(fn(ti))|=n=1|X  fn  dμti,x|< and hence limnfn(ti),gyn(ti)=0 for every i ∈ ℕ. Then λ(X\i=1{ti})=0, i.e., limnfn(t),gyn(t)=0 λ-almost everywhere on X. Since |myn|(A)=Agyn()Edλ for A ∈ 𝓑o, n ∈ ℕ (see (9)) and {|myn|:n} is uniformly λ-continuous, we obtain that {gyn()E:n} is a uniformly Integrable subset of 𝓛1(X). It follows that the family {fn,gyn:n} is also a uniformly integrable subset of 𝓛1 (λ) because supn||fn|| < ∞. Hence using the Vitali theorem, we have that limnXfn,gyndλ=0, and by Corollary 7.11 T is unconditionally converging. □

Remark 7.13: For X being a compact Hausdorff space, the related results to Corollary 7.11 and Theorem 7.12 were obtained by Bombal and Rodrigez-Salinas (see [25, Thorems 1.4 and 1.6]).

Acknowledgement

The author wish to thank the referee for remarks and suggestions that have improved the paper.

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About the article

Received: 2016-03-13

Accepted: 2016-05-31

Published Online: 2016-07-29

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0043. Export Citation

© 2016 Nowak, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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