Results Math Online First c 2017 The Author(s). This article is an open access publication DOI 10.1007/s00025-017-0678-4
Results in Mathematics
Integral Representation of Continuous Operators with Respect to Strict Topologies Marian Nowak Abstract. 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 Eξ of (E, ξ). It is shown that every (β, ξ)-continuous operator T : Cb (X) → E possesses a (β, ξE )-continuous extension Tˆ : B(Bo) → Eξ , where ξE stands for the natural topology on Eξ . 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. Keywords. Spaces of bounded continuous functions, radon measures, vector measures, strict topologies, weakly compact operators, integration operators.
1. 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 Eξ 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
M. Nowak
Results Math
Bo stand for the σ-algebra of Borel sets in X. Let Cb (X) (resp. B(Bo)) be the Banach space of all bounded continuous (resp. bounded Bo-measurable) scalar functions on X, equipped with the topology τu of the uniform norm · ∞ . Let Cb (X) stand for the Banach dual of Cb (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 Cb (X) (resp. B(Bo)) to a lcHs (E, ξ) is said to be weakly compact if T maps τu -bounded sets in Cb (X) (resp., B(Bo)) onto relatively σ(E, Eξ )-compact sets in E. Following [14] the strict topology β on B(Bo) is defined by the family of seminorms pw (v) := sup w(t)|v(t)| for v ∈ B(Bo), t∈X
where w runs over the family Bo (X)+ of all bounded functions w : X → [0, ∞) which vanish at infinity, i.e., for every ε > 0, there is K ∈ K such that supt∈XK w(t) ≤ ε. In view of [14, Theorem 2.4] τc ⊂ β ⊂ τu on B(Bo) and β and τc coincide on any τu -bounded set in B(Bo), where τc denotes the compact-open topology on B(Bo). The topologies β and τu have the same bounded sets. If, in particular, X is compact, then β = τu . The following result characterizes a local base at 0 for β. Proposition 1.1. The strict topology β on B(Bo) has a local base at 0 consisting of all sets of the form: ∞ v ∈ B(Bo) : an sup |v(t)| ≤ 1 , n=1
t∈Kn
where (Kn ) is a sequence of compact sets in X and (an ) is a sequence of positive numbers with lim an = 0. Proof. Assume that wo ∈ Bo (X)+ with wo ∞ = 1. For ε > 0 let Vwo (ε) = {v ∈ B(Bo) : pwo (v) ≤ ε}. Choose a sequence (Kn ) in K such that 1 for n ∈ N. We can assume that ∅ = Ko ⊂ K1 ⊂ supt∈XKn wo (t) ≤ n+1 · · · ⊂ Kn ⊂ · · · . We shall show that ∞ 1 v ∈ B(Bo) : sup |v(t)| ≤ 1 ⊂ Vwo (ε). nε t∈Kn n=1 1 Indeed, assume that v ∈ B(Bo) and nε supt∈Kn |v(t)| ≤ 1 for n ∈ N. Then for t ∈ Kn , wo (t)|v(t)| ≤ max1≤i≤n (supt∈Ki Ki−1 wo (t)|v(t)|) ≤ ε. Since wo (t) = ∞ 0 for t ∈ X n=1 Kn , we have that pwo (v) ≤ ε. Conversely, let (Kn ) be a sequence in K and (an ) be a sequence of positive numbers with lim an = 0. Without loss of generality we can ∞assume that ∅ = K0 ⊂ K1 ⊂ K2 ⊂ · · · ⊂ Kn ⊂ · · · and an ↓ 0. Let wo = n=1 an 1Kn Kn−1 . Then wo ∈ Bo (X)+ and assume that v ∈ Vwo (1), i.e., pwo (v) ≤ 1. Hence for
Integral Representation of Continuous Operators
t ∈ Kn , an|v(t)| ≤ wo (t)|v(t)| ≤ 1, i.e., an supt∈Kn |v(t)| ≤ 1. This means that ∞ Vwo (1) ⊂ n=1 {v ∈ B(Bo) : an supt∈Kn |v(t)| ≤ 1}. The strict topology β restricted to Cb (X) (denoted by β again) has been studied intensively (see [6,7,14,17,33]). Then β can be characterized as the finest locally convex Hausdorff topology on Cb (X) which coincides with τc on τu -bounded sets (see [33, Theorem 2.4], [32]). This means that (Cb (X), β) is a generalized DF-space (see [32, Corollary]); equivalently, β coincides with the mixed topology γ[τu , τc ] in the sense of Wiweger (see [38, (D), pp. 65–66], [7] for more details). If, in particular, X is locally compact, then β coincides with the original strict topology of Buck [4]. If X is compact, we will write simply C(X) instead of Cb (X). For a locally compact Hausdorff space X, Co (X) stands for the Banach space of all those functions of Cb (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 Cb (X), equipped with the strict topology β has the Dunford-Pettis property. The study of continuous linear operators from the Banach space Co (X) to a lcHs has been intensively developed by Edwards [12] and Panchapagesan in a series of papers [26–29] and a monograph [30]. When X is a completely regular Hausdorff space, continuous linear operator from Cb (X), equipped with the different kinds of strict topologies βz (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], Chac`on 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 Cb (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
M. Nowak
Results Math
on Cb (X). Further, we use it to obtain a generalization of a well-known result of Dieudonn´e 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 Cb (X) to (E, ξ) with respect to the representing Borel measures with values in the bidual Eξ of (E, ξ). The strong integrability of functions in B(Bo) is consid , ξ E ) of (E , ξE ), where ξE stands for the ered with respect to the completion (E ξ
ξ
natural topology on Eξ . It is shown that every (β, ξ)-continuous linear operator T : Cb (X) → E possesses a (β, ξE )-continuous extension Tˆ : B(Bo) → Eξ . 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 : Cb (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 c0 , then every (β, ξ)-continuous linear operator T : Cb (X) → E is weakly compact.
2. 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, |μ|(A) = sup {|μ|(K) : K ∈ K, K ⊂ A} = inf {|μ|(O) : O ∈ T , O ⊃ A} . 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, pp. 4–5]): (2.1) sup{|μ(B)| : B ⊂ A, B ∈ Bo} ≤ |μ|(A) ≤ 4 sup{|μ(B)| : B ⊂ A, B ∈ Bo} < ∞.
The following characterization of the topological dual of (Cb (X), β) will be of importance (see [14, Lemma 4.5]). Theorem 2.1. For a linear functional Φ on Cb (X) the following statements are equivalent: (i) Φ is β-continuous.
(ii) There exists a unique μ ∈ M (X) such that Φ(u) = Φμ (u) = X u dμ for u ∈ Cb (X). It is known that for a sequence (un ) in Cb (X), un → 0 in σ(Cb (X), Cb (X)β ) if and only if supn un ∞ < ∞ and un (t) → 0 for every t ∈ X (see [20, Corollary 5]). The following result will be useful (see [25, Lemma 2.5]). Lemma 2.2. Assume that μ ∈ M (X). Then for A ∈ Bo, u dμ : u ∈ Cb (X), u∞ = 1 . |μ|(A) = sup A
Integral Representation of Continuous Operators
In particular, for O ∈ T , we have |μ|(O) = sup u dμ : u ∈ Cb (X), u∞ = 1 and supp u ⊂ O O
and Φμ = |μ|(X). It is known that Cb (X)β is equal to the closure of Cb (X)τc in the Banach space (Cb (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. We will need the following result (see [33, Theorem 5.1]). Theorem 2.3. For a subset M of M (X) the following statements are equivalent: (i) {Φμ : μ ∈ M} is β-equicontinuous. (ii) supμ∈M |μ|(X) < ∞ and {|μ| : μ ∈ M} is uniformly tight, i.e., for every ε > 0 there exists K ∈ K such that supμ∈M |μ|(X K) ≤ ε. 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 Dieudonn´e-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 Cb (X). By τs we denote the topology of simple convergence in ca(Bo). Then τs is generated by the family {pA : A ∈ Bo} of seminorms, where pA (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 supμ∈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., supμ∈M |μ(An )| → 0 whenever An ↓ ∅, (An ) ⊂ Bo. (iii) supμ∈M |μ| (On ) → 0 whenever (On ) is a pairwise disjoint sequence of open sets. (iv) supμ∈M |μ(On )| → 0 whenever (On ) is a pairwise disjoint sequence of open sets. (v) {|μ| : μ ∈ M} is uniformly regular, i.e., for every A ∈ Bo and ε > 0 there exist K ∈ K and O ∈ T with K ⊂ A ⊂ O such that supμ∈M |μ| (O K) ≤ ε. (vi) M is uniformly regular, i.e., for every A ∈ Bo and ε > 0 there exist K ∈ K and O ∈ T with K ⊂ A ⊂ O such that supμ∈M |μ(B)| ≤ ε for every B ∈ Bo with B ⊂ O K.
M. Nowak
Results Math
(vii) supμ∈M | X un dμ| → 0 whenever (un ) is a uniformly bounded sequence in Cb (X) such that un (t) → 0 for every t ∈ X. (viii) supμ∈M | X un dμ| → 0 whenever un → 0 in σ(Cb (X), Cb (X)β ).
(ix) supμ∈M | X un dμ| → 0 whenever (un ) is a uniformly bounded sequence in Cb (X) such that supp un ∩ supp uk = ∅ for n = k. Proof. (i)⇔(ii) It follows from [9, Chap. 7, Theorem 13] because M (X) is a closed subspace of the Banach space ca(Bo), equipped with the norm μ = |μ|(X) (see [18, Chap. 3, § 3, Corollary 3]). (ii)⇒(iii) Assume that (ii) holds. Then supμ∈M |μ|(An ) → 0 for every pairwise disjoint sequence (An ) in Bo (see [9, Theorem 10, pp. 88–89], [10, Proposition 17, p. 8]). Hence (iii) holds. (iii)⇒(iv) It is obvious. (iv)⇒(iii) Assume that (iii) does not hold. Then there exist ε > 0, a sequence (On ) of pairwise disjoint sets in T and a sequence (μn ) in M such that |μn | (On ) ≥ ε for n ∈ N. By (2.1) for every n ∈ N there exists An ∈ Bo with An ⊂ On such that |μn | (On ) ≤ 4|μn (An )| + 4ε . By the regularity of μn , there exists On ∈ T with An ⊂ On such that |μn | (On An ) ≤ 8ε . Let Un = On ∩ On for n ∈ N. Then (Un ) is a pairwise disjoint sequence in T and |μn (An )| ≤ |μn (Un )| + |μn (Un An )| for n ∈ N. Hence |μn (Un An )| ≤ |μn | (Un An ) ≤ |μn | (On An ) ≤ 8ε for n ∈ N, and we get ε ≤ |μn | (On ) ≤ 1 4|μn (Un )| + 34 ε and hence |μn (Un )| ≥ 16 ε. It follows that (iv) does not hold. (iii)⇒(v) Assume that (iii) holds. Then by [T, Theorem 8] the set {|μ| : μ ∈ M} is relatively τs M + (X) -compact in M + (X), where (M + (X) := {μ ∈ M (X) : μ(A) ≥ 0 for every A ∈ Bo}. Hence by [35, Proposition 1] (v) holds. (v)⇔(vi) It follows from (2.1). (v)⇒(ii) It is enough to repeat verbatim the proof of implication (d) ⇒ (e) in [10, Lemma 13, pp. 157–159]. (ii)⇒(vii) Assume that M is uniformly countably additive. Let (un ) be a sequence in Cb (X) such that supn un ∞ = a < ∞ and un (t) → 0 for every t ∈ X. Then there exists λ ∈ ca(Bo)+ such that limλ(A)→0 supμ∈M |μ| (A) = 0. ε Let ε > 0 be given. Then there exists η > 0 such that supμ∈M |μ| (A) ≤ 2(a+1) whenever λ(A) ≤ η, A ∈ Bo. Hence by the Egoroff theorem there exists Aη ∈ ε Bo with λ(X Aη ) ≤ η and supt∈Aη |un (t)| → 0, so supt∈Aη |un (t)| ≤ 2(c+1) for n ≥ no for some no ∈ N, where c = supμ∈M |μ| (X). Then for every μ ∈ M and n ≥ no , we have
un dμ ≤ |un | d|μ| = |un | d|μ| + |un | d|μ| X X Aη XAη ε ε ε |μ| (An ) + a |μ| (X Aη ) ≤ c+a ≤ ε. ≤ 2(a + 1) 2(c + 1) 2(a + 1)
It follows that supμ∈M | X un dμ| ≤ ε for n ≥ no . (vii)⇔(viii) It is obvious. (vii)⇒(ix) It is obvious.
Integral Representation of Continuous Operators
(ix)⇒(iv) Assume that (iv) does not hold. Hence there exist εo > 0, a pairwise disjoint sequence (On ) in T and a sequence (μn ) in M such that |μn (On )| > εo . In view of Lemma 2.2 one can choose a sequence (un ) in Cb (X) with un ∞ = 1 and supp un ⊂ On such that εo εo > . u dμ n n ≥ |μn | (On ) − 2 2 On Then for n ∈ N,
sup un dμ ≥ μ∈M X
εo un dμn > . 2 On
Since supp un ∩ supp uk = ∅ for n = k, we derive that the condition (ix) does not hold. Remark 2.1. The problem of weak compactness of bounded sets of M (X) if X is a locally compact Hausdorff space has been studied by Edwards [12, Theorem 4.22.1] and Panchapagesan [28, Theorems 1 and 2]. In case X is a compact Hausdorff space, an analogous result to Theorem 2.4 can be found in [10, Lemma 13, pp. 157–159] and in [9, Theorem 14, pp. 98–103]. In 1951 Dieudonn´e [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 Dieudonn´e-type theorem in the setting of k-spaces. Corollary 2.5. Assume that X is a k-space and (μn ) is a bounded sequence in the Banach space M (X). Then the following statements are equivalent: (i) μn → μ in σ(M (X), M (X) ) for some μ ∈ M (X). (ii) For every A ∈ Bo, lim μn (A) exists and the set {μn : n ∈ N} is uniformly regular. (iii) For every open set O in X, lim μn (O) exists. Proof. (i)⇒(ii) Assume that (i) holds. Clearly μn (A) → μ(A) for every A ∈ Bo. Since the set {μn : n ∈ N}∪{μ} is σ(M (X), M (X) )-compact, by Theorem 2.4 we obtain that the set {μn : n ∈ N} is uniformly regular. (ii)⇒(iii) It is obvious. (iii)⇒(i) Assume that (iii) holds. Then using Theorem 2.4 and arguing as in the proof of Corollary 4.22.2 of [12] we obtain that (i) holds. Remark 2.2. An analogous result to Corollary 2.5 for X a locally compact Hausdorff space can be found in [12, Corollary 4.22.2] and [29, Corollary 1]. As a consequence of Corollary 2.5 we have:
M. Nowak
Results Math
Corollary 2.6. Assume that X is a k-space. Then the Banach space M (X) is weakly sequentially complete.
3. Integral Representation of Operators on Cb (X) Let Cb (X)β denote the bidual of (Cb (X), β). Since β-bounded subsets of Cb (X) are τu -bounded, the strong topology β(Cb (X)β , Cb (X)) in Cb (X)β coincides with the · -norm topology in Cb (X) restricted to Cb (X)β . Hence Cb (X)β = (Cb (X)β , · ) and Ψ = sup{|Ψ(Φ)| : Φ ∈ Cb (X)β , Φ ≤ 1} for Ψ ∈ Cb (X)β . Then one can embed B(Bo) into Cb (X)β by the mapping π : B(Bo) → Cb (X)β , where for v ∈ B(Bo),
π(v)(Φμ ) := v dμ for μ ∈ M (X). X
We have
|π(v)(Φμ )| = v dμ ≤ v∞ |μ| (X) = v∞ Φμ X
and hence π is bounded and π(v) ≤ v∞ . For t ∈ X let δt (u) := u(t) for u ∈ Cb (X). Then δt ∈ Cb (X)β with δt =
1, and there exists μt ∈ M (X) such that δt (u) = X u dμt for u ∈ Cb (X). Moreover, μt is concentrated on {t}, that is, μt (A) = 1A (t) for A ∈ Bo. Hence, we get π(v) ≥ sup{|π(v)(δt )| : t ∈ X} = sup v dμt : t ∈ X = sup{|v(t)| : t ∈ X} = v∞ .
X
It follows that π(v) = v∞ . Let Eξ be the bidual of (E, ξ), i.e., Eξ = (Eξ , β(Eξ , E)) . Let E denote the family of all ξ-equicontinuous subsets of Eξ . Then ξ is generated by the family of seminorms {pD : D ∈ E}, where pD (e) := sup{|e (e)| : e ∈ D} for e ∈ E. The so-called natural topology ξE on Eξ is generated by the family of seminorms {qD : D ∈ E}, where qD (e ) := sup{|e (e )| : e ∈ D} for e ∈ Eξ (see [12, § 8.7] for more details). Let iE : E → Eξ stand for the canonical injection, that is, iE (e)(e ) = e (e) for e ∈ E and e ∈ Eξ . Let jE : iE (E) → E stand for the left inverse of iE , i.e., jE (iE (e)) = e for e ∈ E. Then iE : E → iE (E) is a (ξ, ξE i (E) )E homeomorphism.
Integral Representation of Continuous Operators
Assume that T : Cb (X) → E is a (β, ξ)-continuous linear operator. Then T is (σ(Cb (X), Cb (X)β ), σ(E, Eξ ))-continuous (see [12, Corollary 8.6.5]) and one can define the conjugate mapping T : Eξ → Cb (X)β by putting T (e ) := e ◦ T for e ∈ Eξ . Then T is (β(Eξ , E), β(Cb (X)β , Cb (X)))-continuous (see [12, Proposition 8.7.1]) and hence T is (σ(Eξ , E), σ(Cb (X)β , Cb (X)))-continuous. It follows that we can define the biconjugate mapping T : Cb (X)β → Eξ by putting T (Ψ)(e ) = Ψ(T (e )) for Ψ ∈ Cb (X)β and e ∈ Eξ . Then T is (σ(Cb (X)β , Cb (X)β ), σ(Eξ , Eξ ))-continuous. Since the topology (τu )E on Cb (X)β coincides with the · -norm topology on Cb (X) restricted to Cb (X)β , in view of [12, Proposition 8.7.2] T is ( · C (X) , ξE )-continuous. Let b
β
Tˆ := T ◦ π : B(Bo) → Eξ . Then Tˆ is a (τu , ξE )-continuous linear operator. For A ∈ Bo let m(A) ˆ := Tˆ(1A ). Hence m ˆ : Bo → Eξ is a finitely additive measure with the ξE -bounded range and is called a representing measure of T . For every e ∈ Eξ , let ˆ ) for A ∈ Bo. m ˆ e (A) := m(A)(e From the general properties of the operator Tˆ it follows immediately that Tˆ(Cb (X)) ⊂ iE (E) and T (u) = jE (Tˆ(u)) for u ∈ Cb (X).
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 ˆ : B ∈ Bo, B ⊂ A} ≤ sup |m ˆ e |(A) sup {qD (m(B)) e ∈D
≤ 4 sup {qD (m(B)) ˆ : B ∈ Bo, B ⊂ A} < ∞.
(3.1)
It follows that supe ∈D |m ˆ e |(X) < ∞ for every D ∈ E. Let v ∈ B(Bo). Choose
ˆ is a a sequence (sn ) in S(Bo) such that sn − v∞ → 0. Then ( X sn dm) ξE -Cauchy sequence in Eξ because for every D ∈ E and k, n ∈ N, we have
sn dm ˆ − sk dm ˆ ≤ sn − sk ∞ · sup |m ˆ e |(X). (3.2) qD X
e ∈D
X
, ξ E ) Then every v ∈ B(Bo) is m-integrable ˆ with respect to the completion (E ξ
of (Eξ , ξE ), and one can define the integral X v dm ˆ by
. v dm ˆ := ξ˜E − lim sn dm ˆ in E (3.3) ξ X
n
X
M. Nowak
Results Math
Definition 3.1. A finitely additive measure m ˆ : Bo → Eξ is said to be ξE -tight if for every D ∈ E and ε > 0 there exists K ∈ K such that qD (m(B)) ˆ = ˆ e (B)| ≤ ε for every B ∈ Bo with B ⊂ X K (equivalently; supe ∈D |m ˆ e |(X K) ≤ ε). supe ∈D |m Now we can state a general Riesz representation theorem for continuous linear operators from (Cb (X), β) to a lcHs (E, ξ). Theorem 3.1. Let T : Cb (X) → E be a (β, ξ)-continuous linear operator and m ˆ be its representing measure. Then the following statements hold: ˆ e ∈ M (X). (i) For every e ∈ Eξ , m
ˆ e for u ∈ Cb (X). (ii) For every e ∈ Eξ , e (T (u)) = X u dm ˆ e ∈ M (X) is (σ(Eξ , E), σ(M (X), Cb (X)))(iii) The mapping Eξ e → m continuous. ˆ e |(X) < ∞ for every D ∈ E) and (iv) m ˆ is ξE -bounded (i.e., supe ∈D |m ξE -tight. (v) For every v ∈ B(Bo), we have
ˆ ˆ T (v) = v dm ˆ and T (v)(e ) = v dm ˆ e for every e ∈ Eξ . X
X
(vi) Tˆ : B(Bo) → Eξ is (β, ξE )-continuous. Conversely, let m ˆ : Bo → Eξ be a finitely additive measure satisfying (i), (iii) and (iv). Then there exists a unique (β, ξ)-continuous linear operator T :
Cb (X) → E such that for every e ∈ Eξ , e (T (u)) = X u dm ˆ e for u ∈ Cb (X). Moreover, m ˆ is the representing measure of T . Proof. (i) Let e ∈ Eξ . Since e ◦ T ∈ Cb (X)β , by Theorem 2.1 there exists a
unique μe ∈ M (X) such that (e ◦T )(u) = X u dμe = Φμe (u) for u ∈ Cb (X). For A ∈ Bo, we have m ˆ e (A) = T (π(1A ))(e ) = π(1A )(T (e )) = π(1A )(e ◦ T ) =
X
1A dμe = μe (A).
It follows that m ˆ e = μe ∈ M (X).
ˆ e for u ∈ Cb (X). (ii) In view of (i) for every e ∈ Eξ , e (T (u)) = X u dm (iii) Since the mapping T : Eξ → Cb (X)β is (σ(Eξ , E), σ(Cb (X)β , Cb (X)))-continuous, the mapping Eξ e → m ˆ e ∈ M (X) is (σ(Eξ , E), σ(M (X), Cb (X)))-continuous. (iv) It follows from (ii) and Theorem 2.3 because for every D ∈ E, the family {e ◦ T : e ∈ D} is β-equicontinuous. (v) Let v ∈ B(Bo). Choose a sequence (sn ) in S(Bo) such that v − sn ∞ → 0. Since Tˆ : B(Bo) → Eξ is (τu , ξE )-continuous, we get
Tˆ(v) = ξE − lim Tˆ(sn ) = ξE − lim sn dm, ˆ (3.4) where
n
s dm ˆ = Tˆ(sn ) ∈ X n
Eξ .
X
Hence using (3.3) we have Tˆ(v) =
X
v dm. ˆ
Integral Representation of Continuous Operators
Let e ∈ Eξ . Define a linear functional ϕe on B(Bo) by ϕe (v) = Tˆ(v)(e ) for v ∈ B(Bo). Note that ϕe = ie ◦ Tˆ, where ie is a linear functional on Eξ defined by ie (e ) = e (e ) for e ∈ Eξ . Since ie is ξE -continuous, we obtain that ϕe is τ u -continuous. Hence there exists a unique μe ∈ ba(Bo) such that ϕe (v) = X v dμe for v ∈ B(Bo) (see [9, Theorem 7, p. 77]). Then for A ∈ Bo, m ˆ e (A) = Tˆ(1A )(e ) = ϕe (1A ) = μe (A), so m ˆ e = μe and Tˆ(v)(e ) = ϕe (v) = X v dm ˆ e for v ∈ B(Bo). (vi) Let D ∈ E and ε > 0 be given. By (iv) there exists a sequence (Kn ) ˆ e |(X Kn ) ≤ 2−2n in K with K1 ⊂ K2 ⊂ · · · ⊂ Kn ⊂ · · · such that supe ∈D |m for n ∈ N. ˆ e |(X)) and η = cεD . Then by Proposition 1.1 Let cD = max(1, supe ∈D |m the set ∞ v ∈ B(Bo) : η −1 2−n+2 sup |v(t)| ≤ 1 U (D, ε) = t∈Kn
n=1
ˆ e |(X is a β-neighborhood of 0 in B(Bo). Note that for e ∈ D, |m 0. Hence for every v ∈ U (D, ε) and e ∈ D, we get |Tˆ(v)(e )| = v dm ˆ e = X
≤
K1
|v| d|m ˆ e | +
≤ η 2−1 cD +
∞
Kn
(Kn+1 Kn )
n=1
∞
∞ n=1
ˆ e |(X) + ≤ sup |v(t)| · |m t∈K1
v dm ˆ e +
∞
n=1
|v| d|m ˆ e | +
∞
n=1
X
Kn+1 Kn
X
Kn ∞
n=1
∞
n=1
Kn ) =
v dm ˆ e
Kn
|v| d|m ˆ e |
|v| d|m ˆ e |
sup |v(t)| · |m ˆ e |(Kn+1 Kn ) ≤
n=1 t∈Kn+1
∞ 1 η 2n−1 · 2−2n ≤ ε. ε+ 2 n=1
It follows that qD (Tˆ(v)) ≤ ε, and this means that Tˆ : B(Bo) → Eξ is (β, ξE )continuous. Conversely, let m ˆ : Bo → Eξ be a finitely additive measure satisfying the conditions (i), (iii) and (iv). For u ∈ Cb (X) define a linear functional ψu on Eξ
by ψu (e ) = X u dm ˆ e for e ∈ Eξ . Then by (iii) ψu is σ(Eξ , E)-continuous, so there is a unique eu ∈ E such that ψu (e ) = e (eu ) for each e ∈ Eξ . For each u ∈ Cb (X) let us put T (u) := eu . Then T : Cb (X) → E is a linear mapping and for every e ∈ Eξ , we have
u dm ˆ e for u ∈ Cb (X). e (T (u)) = e (eu ) = ψu (e ) = 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. Assume that S : Cb (X) → E is another
(β, ξ)-continuous linear operator ˆ e for all u ∈ Cb (X). Then such that for each e ∈ Eξ , e (S(u)) = X u dm e (S(u)) = e (T (u)) for all u ∈ Cb (X), i.e., S = T .
M. Nowak
Results Math
Let m ˆ o be the representing measure of
T . Then by the first part of the ˆ o )e for every u ∈ Cb (X), proof, for each e ∈ Eξ , we get e (T (u)) = X u d(m where (m ˆ o )e ∈ M (X). It follows that (m ˆ o )e = (m) ˆ e for each e ∈ Eξ , i.e., )=m ˆ o (A)(e ) for every A ∈ Bo. Hence m ˆ o = m. ˆ m(A)(e ˆ Remark 3.1. A Riesz representation theorem for bounded linear operators T : C(X) → E, where X is a compact Hausdorff space and E is a Banach space was proved by Bartle, Dunford and Schwartz (see [2, Theorem 3.1], [10, Theorem 1, pp. 152–153]). An analogous theorem for (τu , ξ)-continuous linear operators T : Co (X) → E, where X is a locally compact Hausdorff space and (E, ξ) is a lcHs was proved by Panchapagesan ([27, Theorem 1]). Now we study (β, ξ)-continuous linear operators T : Cb (X) → E such that the set T (D) is relatively σ(Cb (X)β , Cb (X)β )-compact for every D ∈ E (see [12, Corollary 9.3.2]). Proposition 3.2. Let T : Cb (X) → E be a (β, ξ)-continuous linear operator. Then the following statements are equivalent: (i) For every D ∈ E, T (D) is relatively σ(Cb (X)β , Cb (X)β )-compact. (ii) T : Cb (X)β → Eξ is (τ (Cb (X)β , Cb (X)β ), ξE )-continuous. Proof. (i)⇒(ii) Assume that (i) holds and V is a ξE -neighborhood of 0 in Eξ . Then there exists D ∈ E such that D0 ⊂ V , where D0 denotes the polar of D with respect to the dual pair Eξ , Eξ . Then T (D) is a relatively σ(Cb (X)β , Cb (X)β )-compact and by the Krein-Smulian theorem, its closed absolutely convex hull C is still σ(Cb (X)β , Cb (X)β )-compact. Hence C 0 is a τ (Cb (X)β , Cb (X)β )-neighborhood of 0 in Cb (X)β , where C 0 denotes the polar of C with respect to the dual pair Cb (X)β , Cb (X)β . Then T (C 0 ) ⊂ D0 ⊂ V (see [12, § 8.6, (6.6.3)]) and this means that T is (τ (Cb (X)β , Cb (X)β ), ξE )continuous. (ii)⇒(i) Assume that (ii) holds and D ∈ E. It follows that there exists an absolutely convex σ(Cb (X)β , Cb (X)β )-compact set C in Cb (X)β such that T (C 0 ) ⊂ D0 . It follows that T (D) ⊂ C (see [12, § 8.6, (8.6.3)]) and hence T (D) is relatively σ(Cb (X)β , Cb (X)β )-compact. Corollary 3.3. Let T : Cb (X) → E be a (β, ξ)-continuous linear operator. If for every D ∈ E, T (D) is a relatively σ(Cb (X)β , Cb (X)β )-compact subset of Cb (X)β , then the following statements hold: (i) For every v ∈ B(Bo) there exists a net (uα ) in Cb (X) with uα ∞ ≤ v∞ such that Tˆ(uα ) → Tˆ(v) in ξE . (ii) iE (T (Cb (X))) is ξE -dense in Tˆ(B(Bo)). Proof. (i) Let v ∈ B(Bo) and Uv = {u ∈ Cb (X) : u∞ ≤ v∞ }. Then by the Mazur theorem and the Goldstein type theorem (see [31, § 4, Theorem 4,
Integral Representation of Continuous Operators
p. 35]), we get clτ (Cb (X)β ,Cb (X)β ) (π(Uv )) = clσ(Cb (X)β ,Cb (X)β ) (π(Uv )) = {Ψ ∈ Cb (X)β : Ψ ≤ v∞ }. Since π(v) = v∞ , there exists a net (uα ) in Cb (X) with uα ∞ ≤ v∞ such that π(uα ) → π(v) in τ (Cb (X)β , Cb (X)β ). Hence by Proposition 3.2, Tˆ(uα ) = T (π(uα )) → T (π(v)) = Tˆ(v) in ξE . (ii) It follows from (i). Remark 3.2. If X is a compact Hausdorff space, a related result to Corollary 3.3 was obtained by Shuchat [34, Proposition 1]. Definition 3.2 (see [11, § 3]). A finitely additive measure m ˆ : Bo → Eξ is said to be ξE -regular if for every D ∈ E, A ∈ Bo and ε > 0 there exist K ∈ K and O ∈ T with K ⊂ A ⊂ O such that qD (m(B)) ˆ = supe ∈D |m ˆ e (B)| ≤ ε for ˆ e |(O K) ≤ ε). every B ∈ Bo with B ⊂ O K (equivalently, supe ∈D |m Now we present equivalent conditions for a (β, ξ)-continuous linear operator T : Cb (X) → E, the set T (D) to be relatively σ(Cb (X)β , Cb (X)β )compact for every D ∈ E. Corollary 3.4. Assume that X is a k-space. Let T : Cb (X) → E be a (β, ξ)continuous linear operator and m ˆ be its representing measure. Then the following statements are equivalent: (i) For every D ∈ E, T (D) is relatively σ(Cb (X)β , Cb (X)β )-compact. (ii) For every D ∈ E, {m ˆ e : e ∈ D} is relatively σ(M (X), M (X) )compact. (iii) m ˆ : Bo → Eξ is ξE -countably additive. (iv) m ˆ : Bo → Eξ is ξE -regular. (v) T (un ) → 0 in ξ whenever (un ) is a uniformly bounded sequence in Cb (X) such that un (t) → 0 for every t ∈ X. (vi) T (un ) → 0 in ξ whenever (un ) is a uniformly bounded sequence in Cb (X) such that supp un ∩ supp uk = ∅ for n = k. (vii) T (un ) → 0 in ξ whenever un → 0 in σ(Cb (X), Cb (X)β ). ∞
(viii) T (un ) → 0 in ξ whenever n=1 | X un dμ| < ∞ for every μ ∈ M (X). Proof. (i)⇔(ii)⇔(iii)⇔(iv) It follows from Theorem 2.4 because by Theoˆ e |(X) < ∞, where for every e ∈ Eξ , rem 3.1 for every D ∈ E, supe ∈D |m
ˆ e for u ∈ Cb (X). m ˆ e ∈ M (X) and T (e )(u) = (e ◦ T )(u) = X u dm (iii)⇒(v) Assume that m ˆ is ξE -countably additive. Then for every D ∈ E, the family {|m ˆ e | : e ∈ D} is uniformly countably additive and since ˆ e |(X) = c < ∞ (see Theorem 3.1), there exists λ ∈ ca(Bo)+ such supe ∈D |m that {|m ˆ e | : e ∈ D} is uniformly λ-continuous (see [10, Theorem 4, pp. 11– ˆ e |(A) = 0. 12]), i.e., limλ(A)→0 supe ∈D |m Let (un ) be a sequence in Cb (X) such that supn un ∞ = a < ∞ and un (t) → 0 for every t ∈ X. Given ε > 0 there exists η > 0 such that
M. Nowak
Results Math
ε supe ∈D |m ˆ e |(A) ≤ 2(a+1) whenever λ(A) ≤ η, A ∈ Bo. Hence by the Egoroff ε theorem there exists Aη ∈ Bo with λ(X Aη ) ≤ η and supt∈Aη |un (t)| ≤ 2(c+1) for n ≥ nε and some nε ∈ N. Then for every e ∈ D and n ≥ nε , we have
|e (T (un ))| = un dm ˆ e ≤ |un | d|m ˆ e | = |un | d|m ˆ e | + |un | d|m ˆ e | X X Aη XAη ε ε ε ˆ e |(X Aη ) ≤ |m ˆ e |(Aη ) + a|m c+a ≤ ε. ≤ 2(c + 1) 2(c + 1) 2(a + 1)
Hence pD (T (un )) ≤ ε for n ≥ nε and this means that T (un ) → 0 in ξ. (v)⇒(vi) It is obvious. (vi)⇒(ii) Assume that (vi) holds and that (ii) fails to hold, i.e., there ˆ e : e ∈ D0 } is not relatively σ(M (X), M (X) )exists D0 ∈ E such that {m ˆ e |(X) < ∞ (see Theorem 3.1), according to Theocompact. Since supe ∈D |m rem 2.4 there exist ε0 > 0, a pairwise disjoint sequence (On ) of open sets and a ˆ en |(On ) > ε0 . By Lemma 2.2 one can choose sequence (en ) in D0 such that |m a sequence (un ) in Cb (X) with un ∞ = 1 and supp un ⊂ On such that ε0 ε0 ˆ en |(On ) − > . u n dm ˆ en ≥ |m 2 2 On Then for n ∈ N,
pD0 (T (un )) = sup u n dm ˆ e : e ∈ D0 ≥
ε0 u n dm ˆ en > . 2 On
X
On the other hand, since supp un ∩supp uk = ∅ for n = k, we get pD0 (T (un )) → 0. This contradiction establishes that (ii) holds. (v)⇔(vii) It is obvious. (v)⇒(viii) that (v) holds and let (un ) be a sequence in Cb (X)
∞ Assume | u dμ| < ∞ for every μ ∈ M (X). Hence for every μ ∈ such that n=1 X n
M (X), supn | X un dμ| < ∞ and it means that the set {un : n ∈ N} is σ(Cb (X), Cb (X)β )-bounded. Hence {un : n ∈ N} is β-bounded, and it follows that supn un ∞ < ∞. Moreover, since for every t ∈ X, δt ∈ Cb (X)β , we get ∞ ∞ ∞
n=1 |un (t)| = n=1 |δt (un )| = n=1 | X un dμt | < ∞ and it follows that un (t) → 0 for every t ∈ X. Thus T (un ) → 0 in ξ, i.e., (viii) holds. (viii)⇒(vi) Assume that (viii) holds and let (un ) be a sequence in Cb (X) such that supn un ∞ = a < ∞ and supp un ∩ supp uk = ∅ for n = k. Then for every μ ∈ M (X), we have ∞ ∞ ∞ ≤ u dμ |u | d|μ| ≤ a |μ| (supp un ) n n n=1
X
X
n=1 n=1 ∞ = a |μ| supp un ≤ a|μ|(X) < ∞. n=1
Hence T (un ) → 0 in ξ, i.e., (vi) holds.
Integral Representation of Continuous Operators
Assume that T : Cb (X) → E is a (β, ξ)-continuous linear operator such that Tˆ(B(Bo)) = T (π(B(Bo))) ⊂ iE (E). From now on let T (v) := jE (Tˆ(v)) for v ∈ B(Bo). Then T (u) = T (u) for u ∈ Cb (X) and the operator T : B(Bo) → E will be called the natural extension of T on B(Bo). Definition 3.3. A ξ-countably additive measure m : Bo → E is called a ξRadon measure if m is ξ-regular, i.e., for every D ∈ E, A ∈ Bo and ε > 0 there exist K ∈ K and O ∈ T with K ⊂ A ⊂ O such that pD (m(B)) = supe ∈D |me (B)| ≤ ε for every B ∈ Bo with B ⊂ O K (equivalently; supe ∈D |me |(O K) ≤ ε). Theorem 3.5. Assume that X is a k-space and (E, ξ) is a sequentially complete ˆ be lcHs. Let T : Cb (X) → E be a (β, ξ)-continuous linear operator and m its representing measure. Assume that m(A) ˆ ∈ iE (E) for every A ∈ Bo and ˆ Then the following statements hold: m := jE ◦ m. (i) (ii) (iii) (iv)
m : Bo → E is a ξ-Radon measure.
Tˆ(B(Bo)) ⊂ iE (E) and T (v) = X v dm for v ∈ B(Bo). T : B(Bo) → E is (β, ξ)-continuous. T (Cb (X)) is ξ-dense in T (B(Bo)).
ˆ e ∈ M (X), by the Orlicz-Pettis theorem (see [22, Proof. (i) Since me = m Corollary 1]) m is ξ-countably additive, i.e., for every D ∈ E, the family {me : e ∈ D} is uniformly countably additive. Hence by Theorem 2.4 {|me | : e ∈ D} is uniformly regular, i.e., m is ξ-regular. This means that m is a ξ-Radon measure. (ii) Let v ∈ B(Bo).
Choose a sequence (sn ) in S(Bo) such that v − sn ∞ → 0. Note that ( X sn dm) is a ξ-Cauchy sequence in E (see (3.2)) and hence one can define
v dm := ξ − lim sn dm. X
X
Hence using (3.4), we have
iE v dm = ξE − iE sn dm = ξE − sn dm ˆ = v dm ˆ = Tˆ(v). X
X
X
X
It follows that Tˆ(v) ∈ iE (E) and hence T (v) = X v dm. (iii) Since the mapping jE : iE (E) → E is (ξE iE (E) , ξ)-continuous, in view of the condition (vi) of Theorem 3.1, T = jE ◦ Tˆ is (β, ξ)-continuous. (iv) It follows from Corollary 3.3.
M. Nowak
Results Math
4. Weakly Compact Operators on Cb (X) Assume that (E, ξ) is a lcHs. Recall that a (β, ξ)-continuous linear operator T : Cb (X) → E is said to be: ∞ (i) unconditionally convergent ifthe series n=1 T (un ) converges uncondi
∞ tionally in (E, ξ) whenever n=1 | X un dμ| < ∞ for every μ ∈ M (X). (ii) completely continuous if T maps σ(Cb (X), Cb (X)β )-Cauchy sequences in Cb (X) onto ξ-convergent sequences in E. (iii) weakly completely continuous if T maps σ(Cb (X), Cb (X)β )-Cauchy sequences in Cb (X) onto σ(E, Eξ )-convergent sequences in E. Proposition 4.1. If T : Cb (X) → E is a weakly completely continuous operator, then T is unconditionally convergent. Proof. Assume that T weakly
completely continuous. Let (un ) be a sequence is ∞ in Cb (X) such that n=1 | X un dμ| < ∞ for each μ ∈ M (X). For a subsen quence (ukn ) of (un ), let Sn = i=1 uki . Then (Sn ) is a σ(Cb (X), Cb (X)β )∞ Cauchy sequence. It follows that the series n=1 T (ukn ) is σ(E, Eξ )-convergent ∞ in E and in view of the Orlicz-Pettis theorem the series n=1 T (un ) 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 (τu , 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 : Co (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 (τu , ξ)-continuous operator T : Co (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 : Cb (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 : Cb (X) → E be a (β, ξ)-continuous linear operator and m ˆ be its representing measure. Then the following statements are equivalent: (i) (ii) (iii) (iv) (v)
T is weakly compact. T (π(B(Bo))) ⊂ iE (E). m(A) ˆ ∈ iE (E) for every A ∈ Bo. m(O) ˆ ∈ iE (E) for every O ∈ T . For every D ∈ E, the set {m ˆ e : e ∈ D} is relatively σ(M (X), M (X) )compact. (vi) m ˆ : Bo → Eξ is ξE -countably additive. (vii) m ˆ : Bo → Eξ is ξE -regular.
Integral Representation of Continuous Operators
(viii) T (un ) → 0 in ξ whenever (un ) is a uniformly bounded sequence in Cb (X) such that un (t) → 0 for every t ∈ X. (ix) T (un ) → 0 in ξ whenever (un ) is a uniformly bounded sequence in Cb (X) such that supp un ∩ supp uk = ∅ for n = k. (x) T (un ) → 0 in ξ whenever un → 0 in σ(Cb (X), Cb (X)β ). ∞
(xi) T (un ) → 0 in ξ whenever n=1 | X un dμ| < ∞ for every μ ∈ M (X). (xii) T is completely continuous. (xiii) T is unconditionally convergent. (xiv) T is weakly completely continuous. Proof. (i)⇒(ii) Assume that T is weakly compact. Then T (Cb (X)β ) ⊂ iE (E) (see [12, Corollary 9.3.2]) and it follows that T (π(B(Bo))) ⊂ iE (E). (ii)⇒(iii)⇒(iv) It is obvious. (iv)⇒(v) Assume that m(O) ˆ ∈ iE (E) for everyO ∈ T and D ∈ E. Let ∞ (On ) be a pairwise disjoint sequence in T and O = n=1 On . Since for every ˆ e ∈ M (X) (see Theorem 3.1), we get e ∈ Eξ , m e (jE (m(O))) ˆ =m ˆ e (O) =
∞ n=1
m ˆ e (On ) =
∞
e (jE (m(O ˆ n ))).
n=1
Then ˆ = ∞ by the Orlicz-Pettis theorem (see [22, Theorem 1]), we get jE (m(O)) ˆ n )) in (E, ξ). Hence n=1 jE (m(O pD (jE (m(O ˆ n ))) = sup{|e (jE (m(O ˆ n )))| : e ∈ D} = sup{|m ˆ e (On )| : e ∈ D} → 0
and by Theorem 2.4, {m ˆ e : e ∈ D} is relatively σ(M (X), M (X) )-compact. (v)⇔(vi)⇔(vii)⇔(viii) ⇔(ix)⇔(x)⇔(xi) See Corolllary 3.4. (vi)⇒(i) It follows from [12, Corollary 9.3.2]. (iii)⇒(xii) Assume that (iii) holds. Then m = jE ◦ m ˆ : Bo → E is ˆ e ∈ M (X). Assume that (un ) is a ξ-countably additive because me = m σ(Cb (X), Cb (X)β )-Cauchy sequence in Cb (X). Then the set {un : n ∈ N} is β-bounded, so supn un ∞ < ∞. It follows that lim δt (un ) = lim un (t) = vo (t) exists for every t ∈ X, and hence vo ∈ B(Bo). Then by the
Lebesgue bounded convergence theorem (see [27, Proposition 7] T (un ) = X un dm → X vo dm in ξ. (xiii)⇒(xi) It is obvious. (iii)⇒(xiii) Assume that (iii) holds. Then m = jE ◦ m ˆ : Bo →
E is ξ∞ ˆ e ∈ M (X). Assume that n=1 | X un dμ| countable additive because me = m ∞ < ∞ for every μ ∈ M (X). Hence n=1 |un (t)| < ∞ for every t ∈ X because n δt ∈ Cb (X)β . Let Sn (t) = u (t) for t ∈ X. Then (Sn ) is a σ(Cb (X), i i=1 Cb (X)β )-bounded sequence and it follows that (Sn ) is β-bounded and hence supn Sn ∞ < ∞. Let vo (t) = limn Sn (t) for t ∈ X. Then vo ∈ B(Bo) and by the Lebesgue bounded convergence theorem (see [27, Proposition 7]) and Theorem 3.5, we have
M. Nowak
lim n
n i=1
T (ui ) = lim n
X
Results Math
Sn dm =
X
vo dm in (E, ξ).
Finally, if (nj ) is any permutation of N, then limn
∞ t ∈ X. Then j=1 T (unj ) = X vo dm, as desired. (xii)⇒(xiv) It is obvious. (xiv)⇒(xiii) See Proposition 4.1.
n
j=1
unj (t) = vo (t) for
Corollary 4.3. Assume that X is a k-space and (E, ξ) is a quasicomplete lcHs. Let T : Cb (X) → E be a (β, ξ)-continuous weakly compact operator and m ˆ be its representing measure. Then the following statements hold: ˆ : Bo → E is a ξ-Radon measure. (i) m = jE ◦ m (ii) The extension operator T : B(Bo) → E is (β, ξ)-continuous and weakly compact. (iii) T (vn ) → 0 in ξ whenever (vn ) is a uniformly bounded sequence in B(Bo) such that vn (t) → 0 for every t ∈ X. (iv) T (vα ) → 0 in ξ whenever (vα ) is a uniformly bounded net in B(Bo) such that vα → 0 in τc . Proof. (i) It follows from Theorems 4.2 and 3.5. ˆ (ii) By Theorems 4.2 and 3.5 T is (β, ξ)-continuous and since m := jE ◦ m is ξ-countably additive, T is weakly compact (see [26, Theorem 1]). (iii) It follows from the Lebesgue bounded convergence theorem (see [27, Proposition 7]). (iv) It follows from (ii) because β and τc agree on uniformly bounded sets in B(Bo). For definitions and details concerning the strict Dunford-Pettis property, the Dunford-Pettis property and the Dieudonn´e property, we refer a reader to [12, § 9.4]. As an application of Theorem 4.2 we get: Corollary 4.4. Assume that X is a k-space. Then the space (Cb (X), β) has both the strict Dunford-Pettis property and the Dieudonn´e property. Remark 4.1. The fact that the space (Cb (X), β) has the strict Dunford-Pettis property was proved in a different way by Aguayo and Sanchez [1, Theorem 2.4]. Moreover, Khurana [19, Theorem 3] and Chaco´ n and Vielma [5, Theorem 3.1] showed that (Cb (X), β) has the Dunford-Pettis property. Corollary 4.5. Assume that X is a k-space and (E, ξ) is a quasicomplete lcHs. Let T : Cb (X) → E be a (β, ξ)-continuous weakly compact operator. Then (i) T maps relatively σ(Cb (X), M (X))-countably compact sets in Cb (X) onto relatively ξ-compact sets in E. (ii) T maps uniformly bounded relatively τp -sequentially compact sets in Cb (X) onto relatively ξ-compact sets in E (here τp denotes the pointwise convergence topology on Cb (X)).
Integral Representation of Continuous Operators
Proof. (i) Let H be a relatively σ(Cb (X), Cb (X)β )-countably compact subset of Cb (X). Then the closed absolutely convex hull of H is σ(Cb (X), Cb (X)β )-compact (see [H2 , Theorem 4]). Since the space (Cb (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 τp -sequentially compact subset of Cb (X). Then by the Lebesgue dominated convergence theorem, H is relatively σ(Cb (X), Cb (X)β )-sequentially compact (see [H2 , Proposition 2]) and it follows that H is relatively σ(Cb (X), Cb (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 co , then every (τu , ξ)-continuous linear operator T : Co (X) → E is weakly compact (see [29, Corollary 2]). Now we extend this result to operators on Cb (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 co . Then every (β, ξ)-continuous linear operator T : Cb (X) → E is weakly compact. Proof. Let m ˆ be the representing measure of T . Then m ˆ e ∈ M (X) for every e ∈ Eξ (see Theorem 3.1). Assume that (un ) is a sequence in Cb (X) such ∞
that n=1 | X un dμ| < ∞ for every μ ∈ M (X). Hence for every e ∈ Eξ by Theorem 3.1, we have ∞ n=1
∞ |e (T (un ))| = u n dm ˆ e < ∞.
n=1
X
∞ Since E contains no isomorphic copy of co , by [37, Theorem 4] n=1 T (un ) converges unconditionally in (E, ξ) and it means that T is unconditionally convergent. Hence, by Theorem 4.2 T is weakly compact.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
M. Nowak
Results Math
References [1] Aguayo, J., S´ anchez, J.: The Dunford-Pettis property on vector-valued continuous and bounded functions. Bull. Aust. Math. Soc. 48, 303–311 (1993) [2] Bartle, R., Dunford, N., Schwartz, J.: Weak compactness and vector measures. Can. J. Math. 7, 189–305 (1955) [3] Brooks, J.K.: On a theorem of Dieudonn´e. Adv. Math. 36, 165–168 (1980) [4] Buck, R.C.: Bounded contiuous functions on a locally compact space. Mich. Math. J. 5, 95–104 (1958) [5] Chac´ on, G., Vielma, J.: The Dunford-Pettis property and strict topologies. Sci. Math. 2(1), 99–105 (1999) [6] Conway, J.B.: The strict topology and compactness in the space of measures, II. Trans. Am. Math. Soc. 126, 474–486 (1967) [7] Cooper, J.B.: The strict topology and spaces with mixed topologies. Proc. Am. Math. Soc. 30(3), 583–592 (1971) [8] Dieudonn´e, J.: Sur la convergence de suites de mesures de Radon. An. Acad. Brasil. Ci. 23(21–38), 277–282 (1951) [9] Diestel, J.: Sequences and Series in Banach Spaces, Graduate Texts in Math. Springer-Verlag, Berlin (1984) [10] Diestel, J., Uhl, J.J.: Vector measures, Amer. Math. Soc., Math. Surveys 15, Providence, RI, (1977) [11] Dinculeanu, N., Kluvanek, I.: On vector measures. Proc. Lond. Math. Soc. 17(3), 505–512 (1967) [12] Edwards, R.E.: Functional Analysis, Theory and Applications, Holt, Rinehart and Winston, New York (1965) [13] Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989) [14] Giles, R.: A generalization of the strict topology. Trans. Am. Math. Soc. 161, 467–474 (1971) [15] Grothendieck, A.: Sur les applications lineaires faiblement compactnes d’espaces de type C(K). Can. J. Math. 5, 129–173 (1953) [16] Hoffmann-J¨ org¨ensen, J.: Vector measures. Math. Scand. 28, 5–32 (1971) [17] Hoffmann-J¨ org¨ensen, J.: A generalization of the strict topology. Math. Scand. 30, 313–323 (1972) [18] Kantorovitch, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, OxfordElmsford, New York (1982) [19] Khurana, S.S.: Dunford-Pettis property. J. Math. Anal. Appl. 65, 361–364 (1978) [20] Khurana, S.S., Vielma, J.: Weak sequential convergence and weak compactness in spaces of vector-valued continuous functions. J. Math. Anal. Appl. 195, 251– 260 (1995) [21] Lewis, D.R.: Integration with respect to vector measures. Pac. J. Math. 33(1), 157–165 (1970) [22] McArthur, C.: On the theorem of Orlicz and Pettis. Pac. J. Math. 22(2), 297–302 (1967)
Integral Representation of Continuous Operators [23] Nowak, M.: Linear operators on the space of bounded continuous functions with strict topologies. Positivity 14, 831–839 (2010) [24] Nowak, M.: Vector measures and strict topologies. Topol. Appl. 159(5), 1421– 1432 (2012) [25] Nowak, M.: A Riesz representation theory for completely regular Hausdorff spaces and its applications. Open Math. 14, 474–496 (2016) [26] Panchapagesan, T.V.: Applications of a theorem of Grothendieck to vector measures. J. Math. Anal. Appl. 214, 89–101 (1997) [27] Panchapagesan, T.V.: Characterization of weakly compact operators on C0 (T ). Trans. Am. Math. Soc. 350(12), 4849–4867 (1998) [28] Panchapagesan, T.V.: Baire and σ-Borel characterizations of weakly compact sets in M (T ). Trans. Am. Math. Soc. 350(12), 4839–4847 (1998) [29] Panchapagesan, T.V.: Weak compactness of unconditionally convergent operators on C0 (T ). Math. Slovaca 52(1), 57–66 (2002) [30] Panchapagesan, T.V.: The Bartle–Dunford–Schwartz Integral, Mathematical Monographs (New Series), 69. Birkh¨ auser Verlag, Basel (2008) [31] Robertson, A.P., Robertson, W.: Topological Vector Spaces. Cambridge University Press, Cambridge (1964) [32] Schmets, J., Zafarani, J.: Strict topologies and (gDF)-spaces. Arch. Math. 49, 227–231 (1987) [33] Sentilles, F.D.: Bounded continuous functions on a completely regular space. Trans. Am. Math. Soc. 168, 311–336 (1972) [34] Shuchat, A.: Vector measures and scalar operators in locally convex spaces. Mich. Math. J. 24, 303–309 (1977) [35] Siggini, K.K.: Compactness and convergence of set-valued measures. Colloq. Math. 114(2), 177–189 (2009) [36] Topsoe, F.: Compactness in spaces of measures. Stud. Math. 36, 195–212 (1970) [37] Tumarkin, J.B.: Locally convex spaces with basis. Dokl. Acad. Nauk SSSR 195, 1278–1281 (1970). (English Transl., Soviet Math. Dokl. 11 (1970), 1672–1675) [38] Wiweger, A.: Linear spaces with mixed topology. Stud. Math. 20, 47–68 (1961) Marian Nowak Faculty of Mathematics Computer Science and Econometrics University of Zielona G´ ora Ul. Szafrana 4A 65-516 Zielona G´ ora Poland e-mail:
[email protected] Received: December 10, 2016. Accepted: March 24, 2017.
