Abstract
Let \(X\) be a completely regular Hausdorff space and \(C_b(X)\) be the Banach lattice of all real-valued bounded continuous functions on \(X\), endowed with the strict topologies \(\beta _\sigma ,\) \(\beta _\tau \) and \(\beta _t\). Let \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) \((z=\sigma ,\tau ,t)\) stand for the space of all \((\beta _z,\xi )\)-continuous linear operators from \(C_b(X)\) to a locally convex Hausdorff space \((E,\xi ),\) provided with the topology \(\mathcal{T}_s\) of simple convergence. We characterize relative \(\mathcal{T}_s\)-compactness in \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) in terms of the representing Baire vector measures. It is shown that if \((E,\xi )\) is sequentially complete, then the spaces \((\mathcal{L}_{\beta _z,\xi }(C_b(X),E),\mathcal{T}_s)\) are sequentially complete whenever \(z=\sigma \); \(z=\tau \) and \(X\) is paracompact; \(z=t\) and \(X\) is paracompact and Čech complete. Moreover, a Dieudonné–Grothendieck type theorem for operators on \(C_b(X)\) is given.
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1 Introduction and terminology
For terminology concerning vector lattices we refer the reader to [1]. We denote by \(\sigma (L,K),\tau (L,K)\) and \(\beta (L,K)\) the weak topology, the Mackey topology and the strong topology on \(L\), with respect to a dual pair \(\langle L,K\rangle \).
From now on we assume that \(X\) is a completely regular Hausdorff space. Let \(C_b(X)\) be the Banach lattice of all real-valued bounded continuous functions on \(X,\) endowed with the uniform norm \(\Vert \cdot \Vert \). Then the Banach dual \(C_b(X)^{\prime }\) of \(C_b(X)\) with the natural order \((\Phi _1\le \Phi _2\) if \(\Phi _1(u)\le \Phi _2(u)\) for each \(0\le u\in C_b(X))\) is a Dedekind complete Banach lattice. By \(C_b(X)^{\prime \prime }\) we will denote the Banach bidual of \(C_b(X)\).
Let \(\mathcal{B}\) be the algebra of Baire sets in \(X\), which is the algebra generated by the class \(\mathcal{Z}\) of all zero-sets of functions of \(C_b(X)\). Let \(M(X)\) stand for the space of all Baire measures on \(\mathcal{B}\). Then \(M(X)\) with the norm \(\Vert \mu \Vert =|\mu |(X)\) (= the total variation of \(\mu \)) and the natural order \((\mu _1\le \mu _2\) if \(\mu _1(A)\le \mu _2(A)\) for all \(A\in \mathcal{B}\)) is a Dedekind complete Banach lattice (see [20, p. 114, p. 122]). Due to the Alexandrov representation theorem (see [19], [20, Theorem 5.1]) \(C_b(X)^{\prime }\) can be identified with \(M(X)\) through the lattice isomorphism \(M(X)\ni \mu \mapsto \Phi _\mu \in C_b(X)^{\prime }\), where \(\Phi _\mu (u)=\int _X ud\mu \) for all \(u\in C_b(X)\), and \(\Vert \Phi _\mu \Vert =\Vert \mu \Vert \).
The strict topologies \(\beta _\sigma ,\) \(\beta _\tau \) and \(\beta _t\) on \(C_b(X)\) are of importance in the topological measure theory (see [18], [20] for more details). Note that in [18] \(\beta _\sigma ,\beta _\tau ,\beta _t\) are denoted by \(\beta _1,\beta ,\beta _0\) respectively. It is well known that \(\beta _z\) \((z=\sigma ,\tau ,t)\) is a locally convex-solid topology (see [20, Theorem 11.6]), and \(\beta _t\subset \beta _\tau \subset \beta _\sigma \subset \mathcal{T}_{\Vert \cdot \Vert }\). Recall that \(\beta _\sigma \) is a \(\sigma \)-Dini topology (resp. \(\beta _\tau \) is a Dini topology), that is \(u_n\rightarrow 0\) in \(\beta _\sigma \) whenever \(u_n(x)\downarrow 0\) for all \(x\in X\) (resp. \(u_\alpha \rightarrow 0\) in \(\beta _\tau \) whenever \(u_\alpha (x)\downarrow 0\) for all \(x\in X\)) (see [18, Theorem 6.2], [20, Theorems 11.16 and 11.28]). \(\beta _t\) is the finest locally convex topology on \(C_b(X)\) that agrees with the compact-open topology \(\eta \) on each set \(B_r=\{u\in C_b(X),\Vert u\Vert \le r\}\), \(r>0\) (see [20, Theorem 10.5]). Moreover, \((C_b(X),\beta _z)\) (for \(z=\sigma ;\) \(z=\tau \) whevener \(X\) is paracompact; \(z=t\) whenever \(X\) is paracompact and Čech complete) is a strongly Mackey space, that is, every relatively countably \(\sigma (C_b(X)^{\prime }_{\beta _z},C_b(X))\)-compact subset of \(C_b(X)^{\prime }_{\beta _z}\) is \(\beta _z\)-equicontinuous (see [20, Theorems 11.5, 12.22 and 12.9], [18, Theorem 4.5]). We have (see [20, Theorem 11.8], [18, Theorem 4.3]):
where \(M_\sigma (X),\) \(M_\tau (X),\) \(M_t(X)\) are subspaces of \(M(X)\) of all \(\sigma \)-additive \(\tau \)-additive and tight Baire measures, respectively. \(L_\sigma (C_b(X)),\) \(L_\tau (C_b(X))\) and \(L_t(C_b(X))\) are subspaces of \(C_b(X)^{\prime }\) of all \(\sigma \)-additive, \(\tau \)-additive and tight functionals, respectively.
From now on we assume that \((E,\xi )\) is a locally convex Hausdorff space (briefly, lcHs). Let \(\mathcal{P}_\xi \) stand for a directed family of seminorms on \(E\) that generates \(\xi \).
Following the definitions of \(\sigma \)-additive, \(\tau \)-additive and tight functionals on \(C_b(X)\) one can distinguish the corresponding classes of linear operators on \(C_b(X)\).
Definition 1.1
A linear operator \(T:C_b(X)\rightarrow E\) is said to be:
-
(i)
\(\sigma \) -additive if \(T(u_n)\rightarrow 0\) for \(\xi \) whenever \((u_n)\) is a sequence in \(C_b(X)\) such that \(u_n(x)\downarrow 0\) for all \(x\in X\).
-
(ii)
\(\tau \)-additive if \(T(u_\alpha )\rightarrow 0\) for \(\xi \) whenever \((u_\alpha )\) is a net in \(C_b(X)\) such that \(u_\alpha (x)\downarrow 0\) for all \(x\in X\).
-
(iii)
tight if \(T(u_\alpha )\rightarrow 0\) for \(\xi \) whenever \(\sup _\alpha \Vert u_\alpha \Vert <\infty \) and \(u_\alpha \rightarrow 0\) uniformly on compact sets in \(X\).
By \(\mathcal{L}_{\Vert \cdot \Vert ,\xi }(C_b(X),E)\) (resp. \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) for \(z=\sigma ,\tau ,t)\) we will denote the space of all \((\Vert \cdot \Vert ,\xi )\)-continuous (resp. \((\beta _z,\xi )\)-continuous) linear operators \(T:C_b(X)\rightarrow E\). Let \(W(C_b(X),E)\) be the space of all weakly compact operators from the Banach space \(C_b(X)\) to \((E,\xi )\). Then
and
By \(\mathcal{T}_s\) we will denote the topology of simple convergence on \(\mathcal{L}_{\Vert \cdot \Vert ,\xi }(C_b(X),E)\). Then \(\mathcal{T}_s\) is generated by the family \(\{q_{p,u}:p\in \mathcal{P}_\xi ,u\in C_b(X)\}\) of seminorms, where
Graves and Ruess [6, Theorem 7] characterized relative compactness in \(ca(\Sigma ,E)\) \((=\) the space of all \(E\)-valued countably additive measures on a \(\sigma \)-algebra \(\Sigma \)) in the topology \(\mathcal{T}_s\) of simple convergence (convergence on each \(A\in \Sigma \)) in terms of the properties of the integration operators from \(\mathcal{S}(\Sigma )\) to \(E\) and from \(L(\Sigma )\) to \(E\). In [12, Theorem 3.2] (resp. [14, Theorem 3.4]) we study relative \(\mathcal{T}_s\)-compactness in the space \(\mathcal{L}_{\tau ,\xi }(B(\Sigma ),E)\) of all \((\tau (B(\Sigma ),ca(\Sigma )),\xi )\)-continuous linear operators from \(B(\Sigma )\) to \(E\) (resp. in the space \(\mathcal{L}_{\tau ,\xi }(L^\infty (\mu ),E)\) of all \((\tau (L^\infty (\mu ),L^1(\mu )),\xi )\)-continuous linear operators from \(L^\infty (\mu )\) to \(E\)).
In this paper we study topological properties of the spaces \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E),\mathcal{T}_s)\) for \(z=\sigma ,\tau ,t\). We characterize relative \(\mathcal{T}_s\)-compactness in \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) in terms of the corresponding Baire and Borel vector measures (see Theorems 3.2, 4.2, and 5.7 below). It is shown that if \((E,\xi )\) is a sequentially complete lcHs, then the space \((\mathcal{L}_{\beta _z,\xi }(C_b(X),E),\mathcal{T}_s)\) is sequentially complete whenever \(z=\sigma \); \(z=\tau \) and \(X\) is paracompact; \(z=t\) and \(X\) is paracompact and Čech complete (see Corollaries 3.4, 4.5 and 5.4 below). Moreover, we derive a Dieudonné–Grothendieck type theorem for tight and weakly compact operators on \(C_b(X)\) (see Theorem 5.8 below).
2 Representation of continuous operators on \({C_b(X)}\)
Let \(B(\mathcal{B})\) denote the Banach lattice of all functions \(u:X\rightarrow \mathbb R \) that are uniform limits of sequences of \(\mathcal{B}\)-simple functions, provided with the uniform norm \(\Vert \cdot \Vert \).
It is well known that \(C_b(X)\subset B(\mathcal{B})\) (see [2, Lemma 1.2]) and one can embed isometrically \(B(\mathcal{B})\) in \(C_b(X)^{\prime \prime }\) by the mapping \(\pi :B(\mathcal{B})\rightarrow C_b(X)^{\prime \prime }\), where for each \(u\in B(\mathcal{B})\),
Assume that \((E,\xi )\) is a locally convex Hausdorff space. By \((E,\xi )^{\prime }\) or \(E^{\prime }_\xi \) we denote the topological dual of \((E,\xi )\). Then the space \(E^{\prime \prime }_\xi =(E^{\prime }_\xi ,\beta (E^{\prime }_\xi ,E))^{\prime }\) is the bidual of \((E,\xi )\). Let \(\mathcal{E}_\xi \) stand for the set of all \(\xi \)-equicontinuous subsets of \(E^{\prime }_\xi \). Note that \(\xi \) is the topology of uniform convergence on all sets \(\mathcal{A}\in \mathcal{E}_\xi \), i.e., \(\xi \) is generated by the family of seminorms \(\{p_\mathcal{A}:\mathcal{A}\in \mathcal{E}_\xi \}\), where
Let \(\xi _\varepsilon \) stand for the topology on \(E^{\prime \prime }_\xi \) of uniform convergence on all sets \(\mathcal{A}\in \mathcal{E}_\xi \), i.e., \(\xi _\varepsilon \) is generated by the family of seminorms \(\{q_\mathcal{A}:\mathcal{A}\in \mathcal{E}_\xi \}\), where
(see [5, Chapter 8.7]).
Let \(i:E\rightarrow E^{\prime \prime }_\xi \) stand for the canonical embedding, i.e., \(i(e)(e^{\prime })=e^{\prime }(e)\) for \(e\in E\) and \(e^{\prime }\in E^{\prime }_\xi \). Moreover, let \(j:i(E)\rightarrow E\) denote the left inverse of \(i\), that is, \(j\circ i=id_E\). Note that \(j\) is \((\sigma (i(E),E^{\prime }_\xi ),\sigma (E,E^{\prime }_\xi ))\)-continuous.
Assume that \(T:C_b(X)\rightarrow E\) is \((\Vert \cdot \Vert ,\xi )\)-continuous linear operator. Let \(T^{\prime }:E^{\prime }_\xi \rightarrow C_b(X)^{\prime }\) and \(T^{\prime \prime }:C_b(X)^{\prime \prime }\rightarrow E^{\prime \prime }_\xi \) denote the conjugate and the biconjugate of \(T\), respectively. Let
Since the topology \((\mathcal{T}_{\Vert \cdot \Vert _{C_b(X)}})_\varepsilon \) on \(C_b(X)^{\prime \prime }\) coincides with \(\Vert \cdot \Vert _{C_b(X)^{\prime \prime }}\)-topology, in view of [5, Proposition 8.7.2] \(T^{\prime \prime }\) is \((\Vert \cdot \Vert _{C_b(X)^{\prime \prime }},\xi _\varepsilon )\)-continuous. Then \(\hat{T}\) is \((\Vert \cdot \Vert ,\xi _\varepsilon )\)-continuous. For \(A\in \mathcal{B}\) let us put
Then
is a \(\xi _\varepsilon \)-bounded measure, called the representing measure for \(T\). For each \(e^{\prime }\in E^{\prime }_\xi \) let
From the general properties of the operator \(\hat{T}\) it follows immediately that
The next theorem gives a characterization of \((\Vert \cdot \Vert ,\xi )\)-continuous linear operators \(T:C_b(X)\rightarrow E\) in terms of their representing measures (see [13, Theorem 2.1]).
Theorem 2.1
Let \(T:C_b(X)\longrightarrow E\) be a \((\Vert \cdot \Vert ,\xi )\)-continuous linear operator. Then the following statements hold:
-
(i)
\((\hat{m}_T)_{e^{\prime }}\in M(X)\) for each \(e^{\prime }\in E^{\prime }_\xi \).
-
(ii)
The mapping \(E^{\prime }_\xi \ni e^{\prime }\mapsto (\hat{m}_T)_{e^{\prime }}\in M(X)\) is \((\sigma (E^{\prime }_\xi ,E),\sigma (M(X),C_b(X)))\)-continuous.
-
(iii)
For each \(e^{\prime }\in E^{\prime }_\xi ,\)
$$\begin{aligned} \hat{T}(u)(e^{\prime })=e^{\prime }(T(u))=\int \limits _X ud(\hat{m}_T)_{e^{\prime }} \quad \text{ for } \text{ all } \quad u\in C_b(X). \end{aligned}$$
Conversely, let \(\hat{m}:\mathcal{B}\rightarrow E^{\prime \prime }_\xi \) be a vector measure satisfying (i) and (ii). Then there exists a unique \((\Vert \cdot \Vert ,\xi )\)-continuous linear operator \(T:C_b(X)\rightarrow E\) such that (iii) holds and for all \(A\in \mathcal{B}\).
In consequence, the vector measure \(\hat{m}:\mathcal{B}\rightarrow E^{\prime \prime }_\xi \) satisfying (i), (ii) and (iii) is uniquely determined by a \((\Vert \cdot \Vert ,\xi )\)-continuous linear operator \(T:C_b(X)\rightarrow E\).
In view of Theorem 2.1 and (1.1) we have
Corollary 2.2
Let \(T:C_b(X)\rightarrow E\) be a \((\Vert \cdot \Vert ,\xi )\)-continuous linear operator, and \(z=\sigma ,\tau ,t\). Then for each \(e^{\prime }\in E^{\prime }_\xi \) the following statements are equivalent:
-
(i)
\(e^{\prime }\circ T\in C_b(X)^{\prime }_{\beta _z}\).
-
(ii)
\((\hat{m}_T)_{e^{\prime }}\in M_z(X)\).
Note that a subset \(\mathcal{K}\) of \(\mathcal{L}_{\beta _{z,\xi }}(C_b(X),E)\) is \((\beta _z,\xi )\)-equicontinuous \((z=\sigma ,\tau ,t)\) if and only if for each \(\mathcal{A}\in \mathcal{E}_\xi ,\) the set \(\{e^{\prime }\circ T:T\in \mathcal{K},e^{\prime }\in \mathcal{A}\}\) in \(C_b(X)^{\prime }_{\beta _z}\) is \(\beta _z\)-equicontinuous.
The following result will be of importance (see [17, Theorem 2]).
Theorem 2.3
Let \(\mathcal{K}\) be a \(\mathcal{T}_s\)-compact subset of \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) for \(z=\sigma ,\tau ,t\). If \(C\) is a \(\sigma (E^{\prime }_\xi ,E)\)-closed and \(\xi \)-equicontinuous subset of \(E^{\prime }_\xi \), then \(\{e^{\prime }\circ T:T\in \mathcal{K},e^{\prime }\in C\}\) is a \(\sigma (C_b(X)^{\prime }_{\beta _z},C_b(X))\)-compact set in \(C_b(X)^{\prime }_{\beta _z}\).
Assume now that \(T:C_b(X)\rightarrow E\) is a weakly compact operator, that is, \(T\) maps bounded sets in the Banach space \(C_b(X)\) into relatively \(\sigma (E,E^{\prime }_\xi )\)-compact sets in \(E\) (hence \(T\) is \((\Vert \cdot \Vert ,\xi )\)-continuous). Then by the Gantmacher type theorem (see [5, Corollary 9.3.2]) we have
Let us put
and
Note that
Then for each \(e^{\prime }\in E^{\prime }_\xi \) we have
It follows that for each \(\mathcal{A}\in \mathcal{E}_\xi \) and \(A\in \mathcal{B}\) we have
For terminology and basic results concerning integration with respect to vector measures we refer to [7, 10, 15, 16]. Recall that a vector measure \(m:\mathcal{B}\rightarrow E\) is said to be \(\xi \)-strongly bounded if \(m(A_n)\rightarrow 0\) in \(\xi \) for each pairwise disjoint sequence \((A_n)\) in \(\mathcal{B}\).
The following Alexandrov type theorem for weakly compact operators on \(C_b(X)\) is of importance (see [13, Theorems 4.1 and 4.2]).
Theorem 2.4
Assume that \((E,\xi )\) is a quasicomplete lcHs. Then for a weakly compact operator \(T:C_b(X)\rightarrow E\) the following statements hold:
-
(i)
\(m_T:\mathcal{B}\rightarrow E\) is \(\xi \)-strongly bounded.
-
(ii)
\(\hat{m}_T:\mathcal{B}\rightarrow E^{\prime \prime }_\xi \) is \(\xi _\varepsilon \)-strongly bounded.
-
(iii)
\(T(u)=\int _X udm_T\) for all \(u\in C_b(X)\).
3 Topological properties of the space \(\mathcal{L}_{\beta _\sigma ,\xi }(C_b(X),E)\)
We start with a characterization of \((\beta _\sigma ,\xi )\)-equicontinuous sets in \(\mathcal{L}_{\beta _\sigma ,\xi }(C_b(X),E)\).
Proposition 3.1
For a subset \(\mathcal{K}\) of \(\mathcal{L}_{\beta _\sigma ,\xi }(C_b(X),E)\) the following statement are equivalent:
-
(i)
\(\mathcal{K}\) is \((\beta _\sigma ,\xi )\)-equicontinuous.
-
(ii)
\(\mathcal{K}\) is uniformly \(\sigma \)-additive, i.e., \(T(u_n)\rightarrow 0\) in \(\xi \) uniformly for \(T\in \mathcal{K}\) whenever \(u_n(x)\downarrow 0\) for all \(x\in X\).
-
(iii)
The set \(\{\hat{m}_T(A):T\in \mathcal{K},A\in \mathcal{B}\}\) is \(\xi _\varepsilon \)-bounded in \(E^{\prime \prime }_\xi \) and \(\hat{m}_T(Z_n)\rightarrow 0\) in \(\xi _\varepsilon \) uniformly for \(T\in \mathcal{K}\) whenever \(Z_n\downarrow \emptyset \), \(Z_n\in \mathcal{Z}\).
Moreover, if \((E,\xi )\) is a quasicomplete lcHs and \(\mathcal{K}\subset \mathcal{L}_{\beta _\sigma ,\xi }(C_b(X),E)\cap W(C_b(X),E)\), then each of the statements (i)–(iii) is equivalent to the following:
-
(iv)
\(\int _Xu_ndm_T\rightarrow 0\) in \(\xi \) uniformly for \(T\in \mathcal{K}\) whenever \(u_n(x)\downarrow 0\) for \(x\in X\).
-
(v)
The set \(\{m_T(A):T\in \mathcal{K},A\in \mathcal{B}\}\) is \(\xi \)-bounded in \(E\) and \(m_T(Z_n)\rightarrow 0\) in \(\xi \) uniformly for \(T\in \mathcal{K}\) whenever \(Z_n\downarrow \emptyset ,Z_n\in \mathcal{Z}\).
Proof
(i)\(\Longrightarrow \)(ii) Assume that \(\mathcal{K}\) is \((\beta _\sigma ,\xi )\)-equicontinuous. Let \(p\in \mathcal{P}_\xi \) and let \(\varepsilon >0\) be given. Then there is a \(\beta _\sigma \)-neighborhood \(V\) of \(0\) in \(C_b(X)\) such that for each \(T\in \mathcal{K}\) we have \(p(T(u))\le \varepsilon \) for all \(u\in V\). Assume that \((u_n)\) is a sequence in \(C_b(X)\) such that \(u_n(x)\downarrow 0\) for all \(x\in X\). Then \(u_n\rightarrow 0\) for \(\beta _\sigma \) because \(\beta _\sigma \) is a \(\sigma \)-Dini topology. Choose \(n_\varepsilon \in \mathbb N \) such that \(u_n\in V\) for \(n\ge n_\varepsilon \). Hence \(\sup _{T\in \mathcal{K}}p(T(u_n))\le \varepsilon \) for \(n\ge n_\varepsilon \).
(ii)\(\Longrightarrow \)(iii) Assume that \(\mathcal{K}\) is uniformly \(\sigma \)-additive, and let \((u_n)\) be a sequence in \(C_b(X)\) such that \(u_n(x)\downarrow 0\) for all \(x\in X\). Then for each \(\mathcal{A}\in \mathcal{E}_\xi \), we have
This means that the set \(\{e^{\prime }\circ T:T\in \mathcal{K},e^{\prime }\in \mathcal{A}\}\) in \(C_b(X)^{\prime }_{\beta _\sigma }\) is uniformly \(\sigma \)-additive. Assume that \(Z_n\downarrow \emptyset ,\) \(Z_n\in \mathcal{Z}\). In view of [20, Theorem 11.14] we get
This means that \(\hat{m}_T(Z_n)\rightarrow 0\) in \(\xi _\varepsilon \) uniformly for \(T\in \mathcal{K}\). Moreover, we have
It follows that
i.e., the set \(\{\hat{m}_T(A):T\in \mathcal{K},A\in \mathcal{B}\}\) is \(\xi _\varepsilon \)-bounded in \(E^{\prime \prime }_\xi \).
(iii)\(\Longrightarrow \)(i) Assume that \(\{\hat{m}_T(A):T\in \mathcal{K},A\in \mathcal{B}\}\) is \(\xi _\varepsilon \)-bounded in \(E^{\prime \prime }_\xi \) and \(\hat{m}_T(Z_n)\rightarrow 0\) in \(\xi _\varepsilon \) uniformly for \(T\in \mathcal{K}\) whenever \(Z_n\downarrow \emptyset \), \(Z\in \mathcal{Z}\). It follows that for each \(\mathcal{A}\in \mathcal{E}_\xi \), we have
and moreover, for each sequence \((Z_n)\) in \(\mathcal{Z}\) such that \(Z_n\downarrow \emptyset ,\) we have
By [20, Theorem 11.14], we obtain that the set \(\{e^{\prime }\circ T:T\in \mathcal{K},e^{\prime }\in \mathcal{A}\}\) in \(C_b(X)^{\prime }_{\beta _\sigma }\) is \(\beta _\sigma \)-equicontinuous. This means that the set \(\mathcal{K}\) is \((\beta _\sigma ,\xi )\)-equicontinuous.
Assume that \((E,\xi )\) is a quasicomplete lcHs and \(\mathcal{K}\) is a subset of \(\mathcal{L}_{\beta _\sigma ,\xi }(C_b(X),E)\) \(\cap \;W(C_b(X),E)\). Then in view of (2.1) and Theorem 2.4 we obtain that (ii)\(\Longleftrightarrow \)(iv) and (iii)\(\Longleftrightarrow \)(v). \(\square \)
Now we can state a characterization of relatively \(\mathcal{T}_s\)-compact sets in the space \(\mathcal{L}_{\beta _\sigma ,\xi }(C_b(X),E)\).
Theorem 3.2
Let \(\mathcal{K}\) be a subset of \(\mathcal{L}_{\beta _\sigma ,\xi }(C_b(X),E)\). Then the following statements are equivalent:
-
(i)
\(\mathcal{K}\) is relatively \(\mathcal{T}_s\)-compact.
-
(ii)
\(\mathcal{K}\) is \((\beta _\sigma ,\xi )\)-equicontinuous and for each \(u\in C_b(X)\), the set \(\{T(u):T\in \mathcal{K}\}\) is relatively \(\xi \)-compact in \(E\).
-
(iii)
\(\mathcal{K}\) is uniformly \(\sigma \)-additive and for each \(u\in C_b(X)\), the set \(\{T(u):T\in \mathcal{K}\}\) is relatively \(\xi \)-compact in \(E\).
-
(iv)
The following conditions hold:
-
(a)
\(\{\hat{m}_T(A):T\in \mathcal{K},A\in \mathcal{B}\}\) is \(\xi _\varepsilon \)-bounded in \(E^{\prime \prime }_\xi \).
-
(b)
\(\hat{m}_T(Z_n)\rightarrow 0\) in \(\xi _\varepsilon \) uniformly for \(T\in \mathcal{K}\) whenever \(Z_n\downarrow \emptyset \), \(Z_n\in \mathcal{Z}\).
-
(c)
For each \(u\in C_b(X)\), the set \(\{T(u):T\in \mathcal{K}\}\) is relatively \(\xi \)-compact in \(E\).
-
(a)
Proof
(i)\(\Longleftrightarrow \)(ii) See [13, Theorem 3.3].
(ii)\(\Longleftrightarrow \)(iii)\(\Longleftrightarrow \)(iv) It follows from Proposition 3.1. \(\square \)
The following Banach–Steinhaus type theorem for \(\sigma \)-additive operators \(T:C_b(X)\rightarrow E\) will be useful (see [13, Corollary 3.7]).
Proposition 3.3
Let \(T_n:C_b(X)\rightarrow E\) be \(\sigma \)-additive operators for \(n\in \mathbb N \). Assume that \(T(u)=\xi -\lim T_n(u)\) exists for all \(u\in C_b(X)\). Then
-
(i)
\(T:C_b(X)\rightarrow E\) is a \(\sigma \)-additive operator.
-
(ii)
The family \(\{T_n:n\in \mathbb N \}\) is uniformly \(\sigma \)-additive.
As a consequence of Proposition 3.3 we get:
Corollary 3.4
Assume that \((E,\xi )\) is a sequentially complete lcHs. Then the space \((\mathcal{L}_{\beta _\sigma ,\xi }(C_b(X),E),\mathcal{T}_s)\) is sequentially complete.
Proof
Let \((T_n)\) be a \(\mathcal{T}_s\)-Cauchy sequence in \(\mathcal{L}_{\beta _\sigma ,\xi }(C_b(X),E)\). Then for each \(u\in C_b(X),(T_n(u))\) is a \(\xi \)-Cauchy sequence in \(E\), and hence \(T(u)=\xi -\lim T_n(u)\) exists. By Proposition 3.3 the operator \(T:C_b(X)\rightarrow E\) is \(\sigma \)-additive, i.e., \(T\in \mathcal{L}_{\beta _\sigma ,\xi }(C_b(X),E)\) and \(T_n\rightarrow T\) in \(\mathcal{T}_s\), as desired. \(\square \)
4 Topological properties of the space \(\mathcal{L}_{\beta _{\tau },\xi }(C_b(X),E)\)
Now arguing as in the proof of Proposition 3.1 and using [20, Theorem 11.24] and the fact that \(\beta _\tau \) is a Dini topology, we can obtain the following characterization of \((\beta _\tau ,\xi )\)-continuous subsets of \(\mathcal{L}_{\beta _\tau ,\xi }(C_b(X),E)\).
Proposition 4.1
For a subset \(\mathcal{K}\) of \(\mathcal{L}_{\beta _\tau ,\xi } (C_b(X),E)\) the following statements are equivalent:
-
(i)
\(\mathcal{K}\) is \((\beta _\tau ,\xi )\)-equicontinuous.
-
(ii)
\(\mathcal{K}\) is uniformly \(\tau \)-additive, i.e., \(T(u_\alpha )\rightarrow 0\) in \(\xi \) uniformly for \(T\in \mathcal{K}\) whenever \(u_\alpha (x)\downarrow 0\) for all \(x\in X\).
-
(iii)
The set \(\{\hat{m}_T(A):T\in \mathcal{K},A\in \mathcal{B}\}\) is \(\xi _\varepsilon \)-bounded in \(E^{\prime \prime }_\xi \) and \(\hat{m}_T(Z_\alpha )\rightarrow 0\) in \(\xi _\varepsilon \) uniformly for \(T\in \mathcal{K}\) whenever \(Z_\alpha \downarrow \emptyset \), \(Z_\alpha \in \mathcal{Z}\).
Moreover, if \((E,\xi )\) is a quasicomplete lcHs and \(\mathcal{K}\subset \mathcal{L}_{\beta _\tau ,\xi }(C_b(X),E)\cap W(C_b(X),E)\), then each of the statements (i)–(iii) is equivalent to the following:
-
(iv)
\(\int _Xu_\alpha \,dm_T\rightarrow 0\) in \(\xi \) uniformly for \(T\in \mathcal{K}\) whenever \(u_\alpha (x)\downarrow 0\) for \(x\in X\).
-
(v)
The set \(\{m_T(A):T\in \mathcal{K},A\in \mathcal{B}\}\) is \(\xi \)-bounded in \(E\) and \(m_T(Z_\alpha )\rightarrow 0\) in \(\xi \) uniformly for \(T\in \mathcal{K}\) whenever \(Z_\alpha \downarrow \emptyset ,Z_\alpha \in \mathcal{Z}\).
It is known that if \(X\) is paracompact, then \((C_b(X),\beta _\tau )\) is a strongly Mackey space (see [20, Theorem 12.22]). Now we are ready to present a characterization of relatively \(\mathcal{T}_s\)-compact sets in the space \(\mathcal{L}_{\beta _\tau ,\xi }(C_b(X),E)\).
Theorem 4.2
Assume that \(X\) is paracompact. Then for a subset \(\mathcal{K}\) of \(\mathcal{L}_{\beta _\tau ,\xi }(C_b(X),E)\) the following statements are equivalent:
-
(i)
\(\mathcal{K}\) is relatively \(\mathcal{T}_s\)-compact.
-
(ii)
\(\mathcal{K}\) is \((\beta _\tau ,\xi )\)-equicontinuous and for each \(u\in C_b(X)\), the set \(\{T(u):T\in \mathcal{K}\}\) is relatively \(\xi \)-compact in \(E\).
-
(iii)
\(\mathcal{K}\) is uniformly \(\tau \)-additive and for each \(u\in C_b(X)\), the set \(\{T(u):T\in \mathcal{K}\}\) is relatively \(\xi \)-compact in \(E\).
-
(iv)
The following conditions hold:
-
(a)
\(\{\hat{m}_T(A):T\in \mathcal{K},A\in \mathcal{B}\}\) is \(\xi _\varepsilon \)-bounded in \(E^{\prime \prime }_\xi \).
-
(b)
\(\hat{m}_T(Z_\alpha )\rightarrow 0\) in \(\xi _\varepsilon \) uniformly for \(T\in \mathcal{K}\) whenever \(Z_\alpha \downarrow \emptyset \), \(Z_\alpha \in \mathcal{Z}\).
-
(c)
For each \(u\in C_b(X)\), the set \(\{T(u):T\in \mathcal{K}\}\) is relatively \(\xi \)-compact in \(E\).
-
(a)
Proof
(i)\(\Longrightarrow \)(ii) Assume that \(\mathcal{K}\) is relatively \(\mathcal{T}_s\)-compact. Let \(W\) be an absolutely convex and \(\xi \)-closed neighborhood of 0 for \(\xi \) in \(E\). Then the polar \(W^0\) of \(W\) with respect to the dual pair \(\langle E,E^{\prime }_\xi \rangle \) is a \(\sigma (E^{\prime }_\xi ,E)\)-closed and \(\xi \)-equicontinuous subset of \(E^{\prime }_\xi \) (see [1, Theorem 9.21]). Hence in view of Theorem 2.3 the set \(H=\{e^{\prime }\circ T:T\in \mathcal{K},e^{\prime }\in W^0\}\) in \(C_b(X)^{\prime }_{\beta _\tau }\) is relatively \(\sigma (C_b(X)^{\prime }_{\beta _\tau },C_b(X))\)-compact. Since \((C_b(X),\beta _\tau )\) is a strongly Mackey space, the set \(H\) is \(\beta _\tau \)-equicontinuous. It follows that there exists a \(\beta _\tau \)-neighborhood \(V\) of 0 in \(C_b(X)\) such that \(H\subset V^0\), where \(V^0\) is the polar of \(V\) with respect to the dual pair \(\langle C_b(X),C_b(X)^{\prime }_{\beta _\tau }\rangle \). It follows that for each \(T\in \mathcal{K}\) we have that \(\{e^{\prime }\circ T:e^{\prime }\in W^0\} \subset V^0\), i.e., if \(e^{\prime }\in W^0\), then \(|e^{\prime }(T(u))|\le 1\) for all \(u\in V\). This means that for each \(T\in \mathcal{K}\) we have that \(W^0\subset T(V)^0\). Hence \(T(V)\subset T(V)^{00}\subset W^{00}=W\) for each \(T\in \mathcal{K}\), i.e., \(\mathcal{K}\) is \((\beta _\tau ,\xi )\)-equicontinuous. Clearly, for each \(u\in C_b(X)\), the set \(\{T(u):T\in \mathcal{K}\}\) is relatively \(\xi \)-compact in \(E\).
(ii)\(\Longrightarrow \)(i) It follows from [3, Chap. 3, §3.4, Corollary 1].
(ii)\(\Longleftrightarrow \)(iii)\(\Longleftrightarrow \)(iv) It follows from Proposition 4.1. \(\square \)
Now we will need the following result.
Proposition 4.3
Assume that \(X\) is paracompact. Then for a linear operator \(T:C_b(X)\rightarrow E\) the following statements are equivalent:
-
(i)
\(e^{\prime }\circ T\in L_\tau (C_b(X))\) for each \(e^{\prime }\in E^{\prime }_\xi \).
-
(ii)
\(T\) is \((\beta _\tau ,\xi )\)-continuous.
-
(iii)
\(T\) is \(\tau \)-additive.
Proof
(i)\(\Longrightarrow \)(ii) Assume that \(e^{\prime }\circ T\in L_\tau (C_b(X))= C_b(X)^{\prime }_{\beta _\tau }\) for each \(e^{\prime }\in E^{\prime }_\xi \). Then \(T\) is \((\sigma (C_b(X),M_\tau (X)),\sigma (E,E^{\prime }_\xi ))\)-continuous (see [1, Theorem 9.26]). Hence \(T\) is \((\tau (C_b(X),M_\tau (X)),\tau (E,E^{\prime }_\xi ))\)-continuous (see [1, Ex.11, p. 149]). Since \(\beta _\tau =\tau (C_b(X),M_\tau (X))\) (see [20, Theorem 12.22]) and \(\xi \subset \tau (E,E^{\prime }_\xi )\), \(T\) is \((\beta _\tau ,\xi )\)-continuous.
(ii)\(\Longrightarrow \)(iii) Assume that \(T\) is \((\beta _\tau ,\xi )\)-continuous and let \((u_\alpha )\) be a net in \(C_b(X)\) such that \(u_\alpha (x)\downarrow 0\) for all \(x\in X\). Then \(u_\alpha \rightarrow 0\) for \(\beta _\tau \) because \(\beta _\tau \) is a Dini topology. It follows that \(T(u_\alpha )\rightarrow 0\) for \(\xi \).
(iii)\(\Longrightarrow \)(i) It is obvious.\(\square \)
As a consequence of Proposition 4.3 we can derive the following Banach-Steinhaus type theorem for \(\tau \)-additive operators \(T:C_b(X)\rightarrow E\).
Corollary 4.4
Assume that \(X\) is paracompact. Let \(T_n:C_b(X)\rightarrow E\) be \(\tau \)-additive operators for \(n\in \mathbb N \). Assume that \(T(u)=\xi -\lim T_n(u)\) exists for all \(u\in C_b(X)\). Then
-
(i)
\(T\) is a \(\tau \)-additive operator.
-
(ii)
The family \(\{T_n:n\in \mathbb N \}\) is uniformly \(\tau \)-additive.
Proof
For each \(e^{\prime }\in E^{\prime }_\xi \) we have \((e^{\prime }\circ T)(u)= \lim (e^{\prime }\circ T_n)(u)\) for all \(u\in C_b(X)\), and it follows that \((e^{\prime }\circ T_n)\) is a \(\sigma (C_b(X)^{\prime }_{\beta _\tau },C_b(X))\)-Cauchy sequence in \(C_b(X)^{\prime }_{\beta _\tau }\). Since \(X\) is normal and metacompact (see [20, §2]), the space \((C_b(X)^{\prime }_{\beta _\tau },\sigma (C_b(X)^{\prime }_{\beta _\tau },C_b(X)))\) is sequentially complete (see [20, Theorem 14.12], [18, Theorem 8.7], [11]). Hence for \(e^{\prime }\in E^{\prime }_\xi \) there exists \(\Phi _{e^{\prime }}\in C_b(X)^{\prime }_{\beta _\tau }\) such that \(\Phi _{e^{\prime }}(u)=\lim (e^{\prime }\circ T_n)(u)\) for all \(u\in C_b(X)\). It follows that \(e^{\prime }\circ T=\Phi _{e^{\prime }}\in C_b(X)^{\prime }_{\beta _\tau }=L_\tau (C_b(X))\), and by Proposition 4.3 we have that \(T\) is \(\tau \)-additive and \(T_n\rightarrow T\) for \(\mathcal{T}_s\). Since \(\{T_n:n\in \mathbb N \}\cup \{T\}\) is a \(\mathcal{T}_s\)-compact subset of \(\mathcal{L}_{\beta _\tau ,\xi }(C_b(X),E)\), by Theorem 4.2 the set \(\{T_n:n\in \mathbb N \}\) is uniformly \(\tau \)-additive. \(\square \)
Corollary 4.5
Assume that \(X\) is paracompact and \((E,\xi )\) is a sequentially complete lcHs. Then the space \((\mathcal{L}_{\beta _\tau ,\xi }(C_b(X),E),\mathcal{T}_s)\) is sequentially complete.
5 Topological properties of the space \(\mathcal{L}_{\beta _{t},\xi }(C_b(X),E)\)
Recall that \(X\) is said to be Čech complete if it is a \(G_\delta \) subset of its Stone–Čech compactification \(\beta X\) (see [20, §2, p. 106–107]). It is known that if \(X\) is paracompact and Čech complete, then the space \((C_b(X),\beta _t)\) is strongly Mackey (see [20, Theorem 12.9]). Hence using Theorem 2.3 and arguing as in the proof of Theorem 4.2, we can state the following characterization of relatively \(\mathcal{T}_s\)-compact sets in \(\mathcal{L}_{\beta _t,\xi }(C_b(X),E)\).
Theorem 5.1
Assume that \(X\) is paracompact and Čech complete. Then for a subset \(\mathcal{K}\) of \(\mathcal{L}_{\beta _t,\xi }(C_b(X),E)\) the following statements are equivalent:
-
(i)
\(\mathcal{K}\) is relatively \(\mathcal{T}_s\)-compact.
-
(ii)
\(\mathcal{K}\) is \((\beta _t,\xi )\)-equicontinuous and for each \(u\in C_b(X)\), the set \(\{T(u):T\in \mathcal{K}\}\) is relatively \(\xi \)-compact in \(E\).
We will need the following characterization of \((\beta _t,\xi )\)-continuous operators \(T:C_b(X)\rightarrow E\).
Theorem 5.2
Assume that \(X\) is paracompact and Čech complete. Then for a linear operator \(T:C_b(X)\rightarrow E\) the following statements are equivalent:
-
(i)
\(e^{\prime }\circ T\in L_t(C_b(X))\) for each \(e^{\prime }\in E^{\prime }_\xi \).
-
(ii)
\(T\) is \((\beta _t,\xi )\)-continuous.
-
(iii)
\(T\) is tight.
Proof
(i)\(\Longrightarrow \)(ii) Assume that \(e^{\prime }\circ T\in L_t(C_b(X),E)= C_b(X)^{\prime }_{\beta _t}\) for each \(e^{\prime }\in E^{\prime }_\xi \). Then \(T\) is \((\sigma (C_b(X), M_t(X)),\sigma (E,E^{\prime }_\xi ))\)-continuous (see [1, Theorem 9.26]). Hence \(T\) is \((\tau (C_b(X), M_t(X)),\tau (E,E^{\prime }_\xi ))\)-continuous (see [1, Ex. 11, p. 149]). Since \(\beta _t=\tau (C_b(X), M_t(X))\) and \(\xi \subset \tau (E,E^{\prime }_\xi )\), \(T\) is \((\beta _t,\xi )\)-continuous
(ii)\(\Longrightarrow \)(iii) Assume that \(T\) is \((\beta _t,\xi )\)-continuous, and let \((u_\alpha )\) be a net in \(C_b(X)\) such that \(\sup _\alpha \Vert u_\alpha \Vert =r<\infty \) and \(u_\alpha \rightarrow 0\) for the compact-open topology \(\eta \) on \(C_b(X)\). Since \(\eta \big |_{B_r}=\beta _t\big |_{B_r}\) \((B_r=\{u\in C_b(X):\Vert u\Vert \le r\})\), we have that \(u_\alpha \rightarrow 0\) for \(\beta _t\). Hence \(T(u_\alpha )\rightarrow 0\) for \(\xi \).
(iii)\(\Longrightarrow \)(i) It is obvious. \(\square \)
It is known that if \(X\) is paracompact, then \(X\) is metacompact and normal (see [20, §2]). Hence in view of ([20, Theorem 14.12], [11]), we conclude that if \(X\) is paracompact and Čech complete, then the space \((C_b(X)^{\prime }_{\beta _t}, \sigma (C_b(X)^{\prime }_{\beta _t},C_b(X)))\) is sequentially complete. Now we can state the following Banach-Steinhaus type theorem for tight operators \(T:C_b(X)\rightarrow E\).
Corollary 5.3
Assume that \(X\) is paracompact and Čech complete. Let \(T_n:C_b(X)\rightarrow E\) be tight operators for \(n\in \mathbb N \). Assume that \(T(u)=\xi -\lim T_n(u)\) exists for all \(u\in C_b(X)\). Then
-
(i)
\(T\) is a tight operator.
-
(ii)
The family \(\{T_n:n\in \mathbb N \}\) is uniformly tight, i.e., \(T_n(u_\alpha ) \mathop {\longrightarrow }\limits _{\alpha } 0\) in \(\xi \) uniformly for \(n\in \mathbb N \) whenever \(\sup _\alpha \Vert u_\alpha \Vert <\infty \) and \(u_\alpha \rightarrow 0\) uniformly on compact sets in \(X\).
Proof
Arguing as in the Proof of Corollary 4.4 and using Theorem 5.2 we see that \(T:C_b(X)\rightarrow E\) is a tight operator. Since \(\{T_n:n\in \mathbb N \}\cup \{T\}\) is a \(\mathcal{T}_s\)-compact subset of \(\mathcal{L}_{\beta _t,\xi }(C_b(X),E)\), by Theorem 5.1 the family \(\{T_n:n\in \mathbb N \}\) is \((\beta _t,\xi )\)-equicontinuous. Let \(p\in \mathcal{P}_\xi \) and \(\varepsilon >0\) be given. Then there exists a neighborhood \(V\) of 0 for \(\beta _t\) such that \(\sup _n p(T_n(u))\le \varepsilon \) for all \(u\in V\). Assume that \(\sup _\alpha \Vert u_\alpha \Vert <\infty \) and \(u_\alpha \rightarrow 0\) for \(\eta \). Then \(u_\alpha \rightarrow 0\) for \(\beta _t\), and hence there exists \(\alpha _0\) such that \(u_\alpha \in V\) for \(\alpha \ge \alpha _0\). Hence \(\sup _n p(T_n(u_\alpha ))\le \varepsilon \) for \(\alpha \ge \alpha _0\). \(\square \)
Corollary 5.4
Assume that \(X\) is paracompact and Čech complete, and \((E,\xi )\) is a sequentially complete lcHs. Then the space \((\mathcal{L}_{\beta _t,\xi }(C_b(X),E),\mathcal{T}_s)\) is sequentially complete.
Let \(\mathcal{B}a\) (resp. \(\mathcal{B}o\)) denote the \(\sigma \)-algebra of Baire sets (resp. Borel sets) in \(X\). By \(B(\mathcal{B}a)\) (resp. \(B(\mathcal{B}o)\)) we denote the Banach lattice of all bounded \(\mathcal{B}a\)-measurable (resp. \(\mathcal{B}o\)-measurable) functions \(u:X\rightarrow \mathbb R \), provided with the uniform norm \(\Vert \cdot \Vert \).
Let \(m:\mathcal{B}o\rightarrow E\) be a \(\xi \)-countably additive measure. For \(p\in \mathcal{P}_\xi \) we define a semivariation \(\Vert m\Vert _p\) of \(m\) by
where \(V^o_p\) is the polar of \(V_p=\{e\in E:p(e)\le 1\}\) in the duality \(\langle E,E^{\prime }_\xi \rangle \).
We say that \(m\) is inner regular by compact sets (resp. outer regular by open sets) if for each \(A\in \mathcal{B}o\), \(p\in \mathcal{P}_\xi \) and \(\varepsilon >0\) there exists a compact set \(K\) in \(X\), \(K\subset A\) such that \(\Vert m\Vert _p(A\backslash K)\le \varepsilon \) (resp. there exists an open set \(U\) in \(X\), \(A\subset U\) such that \(\Vert m\Vert _p(U\backslash A)\le \varepsilon \)).
Now we present a characterization of tight and weakly compact operators on \(C_b(X)\).
Theorem 5.5
Assume that \((E,\xi )\) is a quasicomplete lcHs. Let \(T:C_b(X)\rightarrow E\) be a weakly compact operator. Then the following statements are equivalent:
-
(i)
\(T\) is \((\beta _t,\xi )\)-continuous.
-
(ii)
\(T\) is tight.
-
(iii)
\(e^{\prime }\circ T\in L_t(C_b(X))\) for each \(e^{\prime }\in E^{\prime }_\xi \).
-
(iv)
\(e^{\prime }\circ m_T\in M_t(X)\) for each \(e^{\prime }\in E^{\prime }_\xi \).
-
(v)
\(m_T\) can be uniquely extended to a \(\xi \)-countably additive Borel measure \(\widetilde{m}_T:\mathcal{B}o\rightarrow E\) which is inner regular by compact sets and outer regular by open sets, and
$$\begin{aligned} T(u)=\int \limits _X u\,dm_T=\int \limits _Xu d\widetilde{m}_T \quad \text{ for } \text{ all } \quad u\in C_b(X). \end{aligned}$$
Proof
(i)\(\Longrightarrow \)(ii) See the proof of implication (i)\(\Longrightarrow \)(ii) of Theorem 5.2.
(ii)\(\Longrightarrow \)(iii)\(\Longrightarrow \)(iv) It is obvious.
(iv)\(\Longrightarrow \)(v) Assume that \(e^{\prime }\circ m_T\in M_t(X)\subset M_\sigma (X)\) for each \(e^{\prime }\in E^{\prime }_\xi \). Since \(m_T\) is \(\xi \)-strongly bounded and \(e^{\prime }\circ m_T:\mathcal{B}\rightarrow E\) is countably additive (see [20, p. 118]), by the Kluvanek Extension Theorem (see [9, Theorem of Extension], [15, Corollary 2]) \(m_T\) can be extended to a \(\xi \)-countably additive measure \(\overline{m}_T:\mathcal{B}a\rightarrow E\), The uniqueness of this extension follows from the uniqueness of the extension of \(e^{\prime }\circ m_T\) from \(\mathcal{B}\) to \(\mathcal{B}a\) for each \(e^{\prime }\in E^{\prime }_\xi \) (see [20, §6, pp. 117-118]).
Hence by [8, Theorem 4] \(\overline{m}_T\) can be uniquely extended to a \(\xi \)-countably additive Borel measure \(\widetilde{m}_T:\mathcal{B}o\rightarrow E\) which is inner regular by compact sets and outer regular by open sets. Since \(C_b(X)\subset B(\mathcal{B})\subset B(\mathcal{B}a)\subset B(\mathcal{B}o)\), we have that
(v)\(\Longrightarrow \)(i) It follows from [8, Theorem 4]. \(\square \)
Now assume that \(T:C_b(X)\rightarrow E\) is a \((\beta _t,\xi )\)-continuous and weakly compact operator. Then by Theorem 5.5, for each \(e^{\prime }\in E^{\prime }_\xi \) we have
for all \(u\in C_b(X)\), where \(\widetilde{e^{\prime }\circ m_T}\) denotes the compact-regular Borel measure that uniquely extends a tight Baire measure \(e^{\prime }\circ m_T\). Hence
Proposition 5.6
Assume that \((E,\xi )\) is a quasicomplete lcHs. For a subset \(\mathcal{K}\) of \(\mathcal{L}_{\beta _t,\xi }(C_b(X),E)\cap W(C_b(X),E)\) the following statements are equivalent:
-
(i)
\(\mathcal{K}\) is \((\beta _t,\xi )\)-equicontinuous.
-
(ii)
The following conditions hold:
-
(a)
\(\sup _{T\in \mathcal{K}}\Vert \widetilde{m}_T\Vert _p(X)<\infty \) for each \(p\in \mathcal{P}_\xi \).
-
(b)
The family \(\{\widetilde{m}_T:T\in \mathcal{K}\}\) of Borel measures is uniformly tight \((\)i.e., for each \(p\in \mathcal{P}_\xi \) and \(\varepsilon >0\) there exists a compact set \(K\) in \(X\) such that \(\sup _{T\in \mathcal{K}}\Vert \widetilde{m}_T\Vert _p(X\backslash K)\le \varepsilon )\).
-
(a)
Proof
(i)\(\Longrightarrow \)(ii) Assume that \(T\) is \((\beta _t,\xi )\)-continuous. Let \(p\in \mathcal{P}_\xi \). Then \(V^o_p\in \mathcal{E}_\xi \) and it follows that the set \(\{e^{\prime }\circ T:T\in \mathcal{K},e^{\prime }\in V^o_p\}\) in \(C_b(X)^{\prime }_{\beta _t}\) is \(\beta _t\)-equicontinuous. Hence in view of (5.1) and (5.2) by [18, Theorem 5.1] we have that
and the family \(\{e^{\prime }\circ \widetilde{m}_T:T\in \mathcal{K},e^{\prime }\in V^o_p\}\) of compact regular scalar Borel measures is uniformly tight, i.e., for each \(\varepsilon >0\) there exists a compact set \(K\) in \(X\) such that \(\sup \{|e^{\prime }\circ \widetilde{m}_T|(X\backslash K): T\in \mathcal{K},e^{\prime }\in V^o_p\}\le \varepsilon \). It follows that \(\sup _{T\in \mathcal{K}}\Vert \widetilde{m}_T\Vert _p (X\backslash K)\le \varepsilon \), as desired.
(ii)\(\Longrightarrow \)(i) Assume that (ii) holds. Then for each \(p\in \mathcal{P}_\xi \) we see that
and the family \(\{e^{\prime }\circ \widetilde{m}_T:T\in \mathcal{K},e^{\prime }\in V^o_p\}\) is uniformly tight. Then by (5.1) and [18, Theorem 5.1], we conclude that the family \(\{e^{\prime }\circ T:T\in \mathcal{K},e^{\prime }\in V^o_p\}\) in \(C_b(X)^{\prime }_{\beta _t}\) is \(\beta _t\)-equicontinuous. It follows that the family \(\mathcal{K}\) is \((\beta _t,\xi )\)-equicontinuous. \(\square \)
As a consequence of Theorem 5.1 and Proposition 5.6 we have:
Theorem 5.7
Assume that \(X\) is Čech complete and paracompact and \((E,\xi )\) is a quasicomplete lcHs. Then for a subset \(\mathcal{K}\) of \(\mathcal{L}_{\beta _t,\xi }(C_b(X),E)\cap W(C_b(X),E)\) the following statements are equivalent:
-
(i)
\(\mathcal{K}\) is relatively \(\mathcal{T}_s\)-compact in \(\mathcal{L}_{\beta _t,\xi }(C_b(X),E)\).
-
(ii)
\(\mathcal{K}\) is \((\beta _t,\xi )\)-equicontinuous and for each \(u\in C_b(X)\), the set \(\{\int _X u d\widetilde{m}_T:T\in \mathcal{K}\}\) is relatively \(\xi \)-compact in \(E\).
-
(iii)
The following conditions hold:
-
(a)
\(\sup _{T\in \mathcal{K}}\Vert \widetilde{m}_T\Vert _p(X)<\infty \) for each \(p\in \mathcal{P}_\xi \).
-
(b)
The family \(\{\widetilde{m}_T:T\in \mathcal{K}\}\) is uniformly tight.
-
(c)
For each \(u\in C_b(X)\), the set \(\{\int _X ud\widetilde{m}_T:T\in \mathcal{K}\}\) is relatively \(\xi \)-compact in \(E\).
-
(a)
Assume that \(X\) is locally compact. Then \(\beta _t\) is the original topology \(\beta \) of Buck (see [4]) and is generated by the family of seminorms \(\{p_v:v\in C_0(X)\}\), where
and \(C_0(X)\) denotes the space of continuous functions on \(X\) vanishing at infinity (see [20, Theorem 10.3] for more details). Then \(\beta _t=\beta _\tau \) (see [20, Theorem 10.14]).
Now we are ready to derive a Dieudonné–Grothendieck type theorem for tight and weakly compact operators on \(C_b(X)\) (see [16, Chapter 5.2]).
Theorem 5.8
Assume that \(X\) is locally compact and \((E,\xi )\) is a quasicomplete lcHs. Let \(T_n:C_b(X)\rightarrow E\) be tight and weakly compact operators for \(n\in \mathbb N \). Assume that \(\xi -\lim \widetilde{m}_{T_n}(A)\) exists for each open Baire set \(A\). Then
-
(i)
\(T(u)=\xi -\lim T_n(u)\) exists for each \(u\in C_b(X)\).
-
(ii)
\(T:C_b(X)\rightarrow E\) is a tight and weakly compact operator.
Proof
In view of [16, Theorem 5.2.23] there exists a unique \(\xi \)-countably additive measure \(\widetilde{m}:\mathcal{B}o\rightarrow E\) which is inner regular by compact sets and outer regular by open sets and such that
for all \(u\in B(\mathcal{B}o)\). Let
Since \(\widetilde{m}\) is \(\xi \)-countably additive, \(\widetilde{m}\) is \(\xi \)-strongly bounded and it follows that the integration operator \(T_{\widetilde{m}}:B(\mathcal{B}o)\rightarrow E\) is weakly compact (see [7, Theorem 7]). Define \(T=T_{\widetilde{m}}\big |_{C_b(X)}:C_b(X)\rightarrow E\). Then \(T\) is weakly compact, and by Theorem 5.5 \(T\) is tight, as desired. \(\square \)
References
Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic Press, New York (1985)
Aguayo, J., Sánchez, J.: Weakly compact operators and the strict topologies. Bull. Aust. Math. Soc. 39, 353–359 (1989)
Bourbaki, N.: Elements of Mathematics, Topological Vector Spaces, Chap. 1–5. Springer, Berlin (1987)
Buck, R.C.: Bounded continuous functions on a locally compact space. Michigan Math. J. 5, 95–104 (1958)
Edwards, R.E.: Functional Analysis, Theory and Applications. Holt, Rinehart and Winston, New York (1965)
Graves, W.H., Ruess, W.: Compactness in spaces of vector-valued measures and a natural Mackey topology for spaces of bounded measurable functions. Contemp. Math. 2, 189–203 (1980)
Hoffmann-Jörgensen, J.: Vector measures. Math. Scand. 28, 5–32 (1971)
Khurana, S.S.: Vector measures on topological spaces. Georgian Math. J. 14(4), 687–698 (2007)
Kluvanek, I., The extension and closure of vector measures. In: Vector and Operator Valued Measures and Applications (Proceedings of Symposium on Snowbird Resort, Alta, Utah, 1972), pp. 175–198. Academic Press, New York (1973)
Lewis, D.R.: Integration with respect to vector measures. Pac. J. Math. 33(1), 157–165 (1970)
Moran, W.: Measures on metacompact spaces. Proc. Lond. Math. Soc. 20, 507–524 (1970)
Nowak, M.: Vector measures and Mackey topologies. Indag. Math. 23, 113–122 (2012)
Nowak, M.: Vector measures and strict topologies. Topol. Appl. 159(5), 1421–1432 (2012)
Nowak, M.: Absolutely continuous on function spaces and vector measures. Positivity. doi:10.1007/s11117-012-0187-3
Panchapagesan, T.V.: Applications of a theorem of Grothendieck to vector measures. J. Math. Anal. Appl. 214, 89–101 (1997)
Panchapagesan, T.V.: The Bartle-Dunford-Schwartz Integral. Monografie Matematyczne, vol. 69. Birkhäuser, Verlag AG (2008)
Schaefer, H., Xiao-Dong, Z.: On the Vitali-Hahn-Saks theorem. In: Operator Theory: Advances and Applications, vol. 75. Birkhäuser, Basel, pp. 289–297 (1995)
Sentilles, F.D.: Bounded continuous functions on a completely regular spaces. Trans. Am. Math. Soc. 168, 311–336 (1972)
Varadarajan, V.S.: Measures on topological spaces. Mat. Sbornik (N.S.), 55:(97), (1961), 35–100; Am. Math. Soc. Transl. 48(2), 161–228 (1965)
Wheeler, R.: A survey of Baire measures and strict topologies. Expositiones Math. 1, 97–190 (1983)
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Nowak, M. Compactness and sequential completeness in some spaces of operators. Positivity 18, 359–373 (2014). https://doi.org/10.1007/s11117-013-0248-2
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DOI: https://doi.org/10.1007/s11117-013-0248-2
Keywords
- Spaces of bounded continuous functions
- Strict topologies
- Dini topologies
- Continuous linear operators
- Topology of simple convergence
- Baire measures
- Banach lattice
- Compactness
- Sequential completeness