Integral Representation of Continuous Operators with Respect to Strict Topologies

Let X be a completely regular Hausdorff space and Bo be the σ-algebra of Borel sets in X. Let Cb(X) (resp. B(Bo)) be the space of all bounded continuous (resp. bounded Bo-measurable) scalar functions on X, equipped with the natural strict topology β. We develop a general integral representation theory of (β, ξ)-continuous operators from Cb(X) to a lcHs (E, ξ) with respect to the representing Borel measure taking values in the bidual E′′ ξ of (E, ξ). It is shown that every (β, ξ)-continuous operator T : Cb(X) → E possesses a (β, ξE)-continuous extension T̂ : B(Bo) → E′′ ξ , where ξE stands for the natural topology on E′′ ξ . If, in particular, X is a k-space and (E, ξ) is quasicomplete, we present equivalent conditions for a (β, ξ)-continuous operator T : Cb(X) → E to be weakly compact. As an application, we have shown that if X is a k-space and a quasicomplete lcHs (E, ξ) contains no isomorphic copy of c0, then every (β, ξ)-continuous operator T : Cb(X) → E is weakly compact. Mathematics Subject Classification. 46G10, 28A32, 28A25, 46A70.


Introduction and Preliminaries
We assume that (E, ξ) is a locally convex Hausdorff space (briefly, lcHs) over either the complex field, C, or the real field, R. By (E, ξ) or E ξ we denote the topological dual of (E, ξ). By σ(L, K), β(L, K) and τ (L, K) we denote weak topology, the strong topology and the Mackey topology on L with respect to a dual pair L, K , respectively.
Throughout the paper we assume that (X, T ) is a completely regular Hausdorff space. By K we will denote the family of all compact sets in X. Let Bo stand for the σ-algebra of Borel sets in X. Let C b (X) (resp. B(Bo)) be the Banach space of all bounded continuous (resp. bounded Bo-measurable) scalar functions on X, equipped with the topology τ u of the uniform norm · ∞ . Let C b (X) stand for the Banach dual of C b (X), equipped with the conjugate norm · . By S(Bo) we denote the space of all Bo-simple functions on X.
Recall that a linear operator T from the Banach space C b (X) (resp. B(Bo)) to a lcHs (E, ξ) is said to be weakly compact if T maps τ u -bounded sets in C b (X) (resp., B(Bo)) onto relatively σ(E, E ξ )-compact sets in E.
Following [14] the strict topology β on B(Bo) is defined by the family of seminorms where w runs over the family B o (X) + of all bounded functions w : X → [0, ∞) which vanish at infinity, i.e., for every ε > 0, there is K ∈ K such that sup t∈X K w(t) ≤ ε.
In view of [14,Theorem 2.4] τ c ⊂ β ⊂ τ u on B(Bo) and β and τ c coincide on any τ u -bounded set in B(Bo), where τ c denotes the compact-open topology on B(Bo). The topologies β and τ u have the same bounded sets. If, in particular, X is compact, then β = τ u .
The following result characterizes a local base at 0 for β. where (K n ) is a sequence of compact sets in X and (a n ) is a sequence of positive numbers with lim a n = 0.
If X is compact, we will write simply C(X) instead of C b (X). For a locally compact Hausdorff space X, C o (X) stands for the Banach space of all those functions of C b (X) that vanish at infinity in X.
The first studies of operators on spaces of scalar continuous functions were made intependently in the fundamental papers of Grothendieck [15] and Bartle, Schwartz and Dunford [2]. In 1953, Grothendieck has showed that there is a one-to-one correspondence between the weakly compact operators from C(X) into a complete lcHs and lcHs-valued Baire measures on a compact Hausdorff space X. But in [15] any theory of integration to represent these operators is not developed. In 1955, Bartle, Dunford and Schwartz [2] developed a theory of integration for scalar functions with respect to Banach space-valued measures and use it to give an integral representation for weakly compact operators T : C(X) → E, where X is a compact Hausdorff space and E is a Banach space. Later, in 1970 Lewis [21] studied a Pettis type weak integral of scalar functions with respect to a countably additive lcHs-valued measure. In particular, in [21] a Bartle-Dunford-Schwartz type theorem for weakly compact operators from C(X) to a lcHs is proved. Moreover, it is showed that if X is a locally compact Hausdorff space, then the space C b (X), equipped with the strict topology β has the Dunford-Pettis property. The study of continuous linear operators from the Banach space C o (X) to a lcHs has been intensively developed by Edwards [12] and Panchapagesan in a series of papers [26][27][28][29] and a monograph [30].
When X is a completely regular Hausdorff space, continuous linear operator from C b (X), equipped with the different kinds of strict topologies β z (z = σ, τ, t, p, g, s), to a lcHs (in particular, a Banach space) have been studied by Lewis [21], Khurana [19], Aguayo and Sanchez [1], Chacòn and Vielma [5] and the present author [23,24].
The aim of this paper is to build a general Riesz representation theory for (β, ξ)-continuous linear operators from C b (X) to an arbitrary lcHs (E, ξ), extending a number of results which are generally known to be true if X is a compact Hausdorff space and E is a Banach space. In Sect. 2, making use of the results of Topsoe [36] we state characterizations of weak compactness of bounded sets in the Banach space M (X) of scalar Radon measures in case X is a k-space. These characterizations play a key role in the studies of operators on C b (X). Further, we use it to obtain a generalization of a well-known result of Dieudonné for the weak sequential convergence in M (X). In Sect. 3 we study the problem of an integral representation for (β, ξ)-continuous operators from C b (X) to (E, ξ) with respect to the representing Borel measures with values in the bidual E ξ of (E, ξ). The strong integrability of functions in B(Bo) is considered with respect to the completion ( E ξ , ξ E ) of (E ξ , ξ E ), where ξ E stands for the natural topology on E ξ . It is shown that every (β, ξ)-continuous linear opera- In Sect. 4, if X is a k-space and (E, ξ) is quasicomplete, we present equivalent conditions for a (β, ξ)-continuous linear operator T : C b (X) → E to be weakly compact, in particular, in terms of the representing measures. As a consequence, we derive that if X is a k-space and (E, ξ) is quasicomplete and contains no isomorphic copy of c 0 , then every (β, ξ)-continuous linear operator T : C b (X) → E is weakly compact.

Topological Properties of Spaces of Scalar Measures
Recall that a countably additive scalar measure μ on Bo is called a Radon measure if its variation |μ| is regular, i.e., for each A ∈ Bo, Let M (X) denote the space of all Radon measures, equipped with the total variation norm μ := |μ|(X). Note that for μ ∈ M (X) and A ∈ Bo (see [10,Proposition 11,): The following characterization of the topological dual of (C b (X), β) will be of importance (see [14,Lemma 4.5]). Theorem 2.1. For a linear functional Φ on C b (X) the following statements are equivalent: It is known that for a sequence ( and only if sup n u n ∞ < ∞ and u n (t) → 0 for every t ∈ X (see [20,Corollary 5]).

Lemma 2.2.
Assume that μ ∈ M (X). Then for A ∈ Bo, In particular, for O ∈ T , we have It is known that C b (X) β is equal to the closure of C b (X) τc in the Banach space (C b (X) , · ) (see [7,Proposition 1]). Hence in view of Lemma 2.2, the space M (X) (equipped with the total variation norm μ = |μ|(X)) is a Banach space.
Recall that a completely regular Hausdorff space (X, T ) is a k-space if a subset A of X is closed whenever A ∩ K is compact for all compact sets in X. In particular, every locally compact Hausdorff space, every metrizable space and every space satisfying the first countability axiom is a k-space (see [13, Chap. 3, § 3]). Now using the results of Topsoe's paper [36] 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 [15,Theorem 2], [9, Theorem 14, pp. 98-103]), which plays a crucial role in the study of operators on C b (X).
By τ s we denote the topology of simple convergence in ca(Bo). Then τ s is generated by the family {p A : A ∈ Bo} of seminorms, where p A (μ) = |μ(A)| for μ ∈ ca(Bo).

Theorem 2.4.
Assume that X is a k-space and M is a subset of M (X) such that sup μ∈M |μ|(X) < ∞. Then the following statements are equivalent: (i) M is relatively weakly compact in the Banach space M (X).
(ii) M is uniformly countably additive, i.e., sup μ∈M |μ(A n )| → 0 whenever Proof. (iii)⇒(iv) It is obvious. (ii)⇒(vii) Assume that M is uniformly countably additive. Let (u n ) be a sequence in C b (X) such that sup n u n ∞ = a < ∞ and u n (t) → 0 for every t ∈ X. Then there exists λ ∈ ca(Bo) + such that lim λ(A)→0 sup μ∈M |μ| (A) = 0. Let ε > 0 be given. Then there exists η > 0 such that sup μ∈M |μ| (A) ≤ ε Then for n ∈ N, Since supp u n ∩ supp u k = ∅ for n = k, we derive that the condition (ix) does not hold. In 1951 Dieudonné [8] proved that, if a sequence of Radon measures defined on the Borel σ-algebra of a compact metrizable space converges on every open set, then it converges on every Borel set, and in this case the sequence is uniformly regular. Brooks [3] generalizes this theorem to the case the space is either compact or the space is normal and the sequence is uniformly bounded.
As an application of Theorem 2.4 we can state a Dieudonné-type theorem in the setting of k-spaces.

Corollary 2.5.
Assume that X is a k-space and (μ n ) is a bounded sequence in the Banach space M (X). Then the following statements are equivalent: (ii) For every A ∈ Bo, lim μ n (A) exists and the set {μ n : n ∈ N} is uniformly regular.

Integral Representation of Operators on
We have Let E ξ be the bidual of (E, ξ), i.e., E ξ = (E ξ , β(E ξ , E)) . Let E denote the family of all ξ-equicontinuous subsets of E ξ . Then ξ is generated by the family of seminorms The so-called natural topology ξ E on E ξ is generated by the family of seminorms {q D : D ∈ E}, where q D (e ) := sup{|e (e )| : e ∈ D} for e ∈ E ξ (see [12, § 8.7] for more details).
Let i E : E → E ξ stand for the canonical injection, that is, i E (e)(e ) = e (e) for e ∈ E and e ∈ E ξ . Let j E : i E (E) → E stand for the left inverse of i E , i.e., j E (i E (e)) = e for e ∈ E. Then i E : [12,Corollary 8.6.5]) and one can define the conjugate mapping )-continuous (see [12,Proposition 8.7.1]) and hence T is (σ(E ξ , E), σ(C b (X) β , C b (X)))-continuous. It follows that we can define the biconjugate mapping ThenT is a (τ u , ξ E )-continuous linear operator. From the general properties of the operatorT it follows immediately that From now on we will use the integration theory of scalar functions with respect to vector measures that is developed in [16,26,30].
In view of (2.1) for D ∈ E and A ∈ Bo, we have Then every v ∈ B(Bo) ism-integrable with respect to the completion ( E ξ , ξ E ) of (E ξ , ξ E ), and one can define the integral X v dm by

3)
Definition 3.1. A finitely additive measurem : Bo → E ξ is said to be ξ E -tight if for every D ∈ E and ε > 0 there exists K ∈ K such that q D (m(B)) = sup e ∈D |m e (B)| ≤ ε for every B ∈ Bo with B ⊂ X K (equivalently; sup e ∈D |m e |(X K) ≤ ε). Now we can state a general Riesz representation theorem for continuous linear operators from (C b (X), β) to a lcHs (E, ξ). E be a (β, ξ)-continuous linear operator and m be its representing measure. Then the following statements hold: Conversely, letm : Bo → E ξ be a finitely additive measure satisfying (i), (iii) and (iv). Then there exists a unique (β, ξ)-continuous linear operator T : It follows thatm e = μ e ∈ M (X).
(ii) In view of (i) for every e ∈ E ξ , e (T (u)) = X u dm e for u ∈ C b (X).
(iii) Since the mapping T : (iv) It follows from (ii) and Theorem 2.3 because for every D ∈ E, the family {e • T : e ∈ D} is β-equicontinuous. ( where X s n dm =T (s n ) ∈ E ξ . Hence using (3.3) we haveT (v) = X v dm.
Let e ∈ E ξ . Define a linear functional ϕ e on B(Bo) by ϕ e (v) =T (v)(e ) for v ∈ B(Bo). Note that ϕ e = i e •T , where i e is a linear functional on E ξ defined by i e (e ) = e (e ) for e ∈ E ξ . Since i e is ξ E -continuous, we obtain that ϕ e is τ u -continuous. Hence there exists a unique μ e ∈ ba(Bo) such that ϕ e (v) = X v dμ e for v ∈ B(Bo) (see [ (vi) Let D ∈ E and ε > 0 be given. By (iv) there exists a sequence (K n ) in K with K 1 ⊂ K 2 ⊂ · · · ⊂ K n ⊂ · · · such that sup e ∈D |m e |(X K n ) ≤ 2 −2n for n ∈ N.
Let c D = max(1, sup e ∈D |m e |(X)) and η = ε cD . Then by Proposition 1.1 the set It follows that q D (T (v)) ≤ ε, and this means thatT : B(Bo) → E ξ is (β, ξ E )continuous. Conversely, letm : Bo → E ξ be a finitely additive measure satisfying the conditions (i), (iii) and (iv). For u ∈ C b (X) define a linear functional ψ u on E ξ by ψ u (e ) = X u dm e for e ∈ E ξ . Then by (iii) ψ u is σ(E ξ , E)-continuous, so there is a unique e u ∈ E such that ψ u (e ) = e (e u ) for each e ∈ E ξ . For each u ∈ C b (X) let us put T (u) := e u . Then T : C b (X) → E is a linear mapping and for every e ∈ E ξ , we have e (T (u)) = e (e u ) = ψ u (e ) = X u dm e for u ∈ C b (X).
In view of (i) and (iv) and Theorem 2.3 for every D ∈ E, the family {e • T : e ∈ D} is β-equicontinuous and it follows that T is (β, ξ)-continuous.
Then the following statements are equivalent: Proof. (i)⇒(ii) Assume that (i) holds and V is a ξ E -neighborhood of 0 in E ξ . Then there exists D ∈ E such that D 0 ⊂ V , where D 0 denotes the polar of D with respect to the dual pair E ξ , E ξ . Then T (D) is a relatively σ(C b (X) β , C b (X) β )-compact and by the Krein-Smulian theorem, its closed absolutely convex hull C is still [12, § 8.6, (6.6

.3)]) and hence
then the following statements hold:   E be a (β, ξ)continuous linear operator andm be its representing measure. Then the following statements are equivalent: Proof. (i)⇔(ii)⇔(iii)⇔(iv) It follows from Theorem 2.4 because by Theorem 3.1 for every D ∈ E, sup e ∈D |m e |(X) < ∞, where for every e ∈ E ξ , m e ∈ M (X) and T (e )(u) = (e • T )(u) = X u dm e for u ∈ C b (X).
Let (u n ) be a sequence in C b (X) such that sup n u n ∞ = a < ∞ and u n (t) → 0 for every t ∈ X. Given ε > 0 there exists η > 0 such that sup e ∈D |m e |(A) ≤ ε 2(a+1) whenever λ(A) ≤ η, A ∈ Bo. Hence by the Egoroff theorem there exists A η ∈ Bo with λ(X A η ) ≤ η and sup t∈Aη |u n (t)| ≤ ε 2(c+1) for n ≥ n ε and some n ε ∈ N. Then for every e ∈ D and n ≥ n ε , we have Hence p D (T (u n )) ≤ ε for n ≥ n ε and this means that T (u n ) → 0 in ξ.
Then for n ∈ N, On the other hand, since supp u n ∩supp u k = ∅ for n = k, we get p D0 (T (u n )) → 0. This contradiction establishes that (ii) holds.
(viii)⇒(vi) Assume that (viii) holds and let (u n ) be a sequence in C b (X) such that sup n u n ∞ = a < ∞ and supp u n ∩ supp u k = ∅ for n = k. Then for every μ ∈ M (X), we have Hence T (u n ) → 0 in ξ, i.e., (vi) holds.

Weakly Compact Operators on C b (X)
Assume that (E, ξ) is a lcHs. Recall that a (β, ξ)-continuous linear operator T : C b (X) → E is said to be: Proof. Assume that T is weakly completely continuous. Let (u n ) be a sequence in E and in view of the Orlicz-Pettis theorem the series ∞ n=1 T (u n ) is unconditionally converging (see [22,Theorem 1]). This means that T is unconditionally convergent.
When X is a compact Hausdorff space and (E, ξ) is a complete lcHs, Grothendieck [15,Theorem 6] gave some necessary and sufficient conditions for (τ u , ξ)-continuous operator T : C(X) → E to be weakly compact. Later, Edwards [12,Theorem 9.4.10] obtained characterizations of weakly compact operators T : C o (X) → E, where X is a locally compact Hausdorff space and (E, ξ) is complete. Panchapagesan (see [27,Theorems 2,3 and 12], [30,Theorem 5.3.7]) has presented equivalent conditions for a (τ u , ξ)-continuous operator T : C o (X) → E to be weakly compact if X is a locally compact Hausdorff space and (E, ξ) is quasicomplete. Now using the results of Sect. 3 we present equivalent conditions for a (β, ξ)-continuous operator T : C b (X) → E to be weakly compact, where X is a k-space and (E, ξ) is quasicomplete.

Theorem 4.2.
Assume that X is a k-space and (E, ξ) is a quasicomplete lcHs. Let T : C b (X) → E be a (β, ξ)-continuous linear operator andm be its representing measure. Then the following statements are equivalent: [12,Corollary 9.3.2]) and it follows that T (π(B(Bo))) ⊂ i E (E).
(ii)⇒(iii)⇒(iv) It is obvious. (iii)⇒(xii) Assume that (iii) holds. Then m = j E •m : Bo → E is ξ-countably additive because m e =m e ∈ M (X). Assume that (u n ) is a σ(C b (X), C b (X) β )-Cauchy sequence in C b (X). Then the set {u n : n ∈ N} is β-bounded, so sup n u n ∞ < ∞. It follows that lim δ t (u n ) = lim u n (t) = v o (t) exists for every t ∈ X, and hence v o ∈ B(Bo). Then by the Lebesgue bounded convergence theorem (see [27,Proposition 7] -bounded sequence and it follows that (S n ) is β-bounded and hence sup n S n ∞ < ∞. Let v o (t) = lim n S n (t) for t ∈ X. Then v o ∈ B(Bo) and by the Lebesgue bounded convergence theorem (see [27,Proposition 7]) and Theorem 3.5, we have Finally, if (n j ) is any permutation of N, then lim n n j=1 u nj (t) = v o (t) for t ∈ X. Then ∞ j=1 T (u nj ) = X v o dm, as desired. (xii)⇒(xiv) It is obvious. (xiv)⇒(xiii) See Proposition 4.1. X is a k-space and (E, ξ) is a quasicomplete lcHs. Let T : C b (X) → E be a (β, ξ)-continuous weakly compact operator andm be its representing measure. Then the following statements hold:
Proof. (i) It follows from Theorems 4.2 and 3.5.
For definitions and details concerning the strict Dunford-Pettis property, the Dunford-Pettis property and the Dieudonné property, we refer a reader to [12, § 9.4]. As an application of Theorem 4.2 we get:   Assume that X is a k-space and (E, ξ) is a quasicomplete lcHs. Let T : C b (X) → E be a (β, ξ)-continuous weakly compact operator. Then (i) T maps relatively σ(C b (X), M(X))-countably compact sets in C b (X) onto relatively ξ-compact sets in E. (ii) T maps uniformly bounded relatively τ p -sequentially compact sets in C b (X) onto relatively ξ-compact sets in E (here τ p denotes the pointwise convergence topology on C b (X)).
Proof. (i) Let H be a relatively σ(C b (X), C b (X) β )-countably compact subset of C b (X). Then the closed absolutely convex hull of H is σ(C b (X), C b (X) β )-compact (see [H 2 , Theorem 4]). Since the space (C b (X), β) has the Dunford-Pettis property (see [19,Theorem 3]), we obtain that T (H) is relatively ξ-compact. (ii) Let H be a uniformly bounded relatively τ p -sequentially compact subset of C b (X). Then by the Lebesgue dominated convergence theorem, H is relatively σ(C b (X), C b (X) β )-sequentially compact (see [H 2 , Proposition 2]) and it follows that H is relatively σ(C b (X), C b (X) β )-countably compact. Hence in view of (i) T (H) is relatively ξ-compact.
Grothendieck [15] proved that if X is a compact Hausdorff space and E is a weakly sequentially complete Banach space, then every bounded operator T : C(X) → E is weakly compact. Panchapagesan has shown that if X is a locally compact Hausdorff space and (E, ξ) is quasicomplete and contains no isomorphic copy of c o , then every (τ u , ξ)-continuous linear operator T : C o (X) → E is weakly compact (see [29,Corollary 2]). Now we extend this result to operators on C b (X) in the setting of k-spaces. Corollary 4.6. Assume that X is a k-space and (E, ξ) is a quasicomplete lcHs that contains no isomorphic copy of c o . Then every (β, ξ)-continuous linear operator T : C b (X) → E is weakly compact.
Proof. Letm be the representing measure of T . Thenm e ∈ M (X) for every e ∈ E ξ (see Theorem 3.1). Assume that (u n ) is a sequence in C b (X) such that ∞ n=1 | X u n dμ| < ∞ for every μ ∈ M (X). Hence for every e ∈ E ξ by Theorem 3.1, we have ∞ n=1 |e (T (u n ))| = ∞ n=1 X u n dm e < ∞.
Since E contains no isomorphic copy of c o , by [37,Theorem 4] ∞ n=1 T (u n ) converges unconditionally in (E, ξ) and it means that T is unconditionally convergent. Hence, by Theorem 4.2 T is weakly compact.
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