Compactness and sequential completeness in some spaces of operators

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.

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]):

$$\begin{aligned} (C_b(X),\beta _z)^{\prime }=\{\Phi _\mu :\mu \in M_z(X)\}=L_z(C_b(X)) \ \ (z=\sigma ,\tau ,t), \end{aligned}$$

(1.1)

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

$$\begin{aligned} \mathcal{L}_{\beta _t,\xi }(C_b(X),E)\subset \mathcal{L}_{\beta _\tau ,\xi }(C_b(X),E)\subset \mathcal{L}_{\beta _\sigma ,\xi }(C_b(X),E)\subset \mathcal{L}_{\Vert \cdot \Vert ,\xi }(C_b(X),E) \end{aligned}$$

and

$$\begin{aligned} W(C_b(X),E)\subset \;\mathcal{L}_{\Vert \cdot \Vert ,\xi }(C_b(X),E). \end{aligned}$$

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

$$\begin{aligned} q_{p,u}(T):=p(T(u)) \ \ for \ \ T\in \mathcal{L}_{\Vert \cdot \Vert ,\xi }(C_b(X),E). \end{aligned}$$

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 $u\in B(\mathcal{B})$,

$$\begin{aligned} \pi (u)(\Phi _\mu )=\int \limits _X ud\mu \quad \text{ for } \text{ all } \quad \mu \in M(X). \end{aligned}$$

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

$$\begin{aligned} p_\mathcal{A}(e)=\sup \{|e^{\prime }(e)|:e^{\prime }\in \mathcal{A}\} \quad \text{ for } \quad e\in E. \end{aligned}$$

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

$$\begin{aligned} q_\mathcal{A}(e^{\prime \prime })=\sup \{|e^{\prime \prime }(e^{\prime })|:e^{\prime }\in \mathcal{A}\} \quad \text{ for } \quad e^{\prime \prime }\in E^{\prime \prime }_\xi , \end{aligned}$$

(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

$$\begin{aligned} \hat{T}:=T^{\prime \prime }\circ \pi :B(\mathcal{B})\rightarrow E^{\prime \prime }_\xi . \end{aligned}$$

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

$$\begin{aligned} \hat{m}_T:\mathcal{B}\longrightarrow E^{\prime \prime }_\xi \end{aligned}$$

is a $\xi _\varepsilon$-bounded measure, called the representing measure for $T$. For each $e^{\prime }\in E^{\prime }_\xi$ let

$$\begin{aligned} (\hat{m}_T)_{e^{\prime }}(A):=\hat{m}_T(A)(e^{\prime }) \quad \text{ for } \text{ all } \quad A\in \mathcal{B}. \end{aligned}$$

From the general properties of the operator $\hat{T}$ it follows immediately that

$$\begin{aligned} \hat{T}(C_b(X))\subset i(E) \quad \text{ and } \quad T(u)=j(\hat{T}(u)) \quad \text{ for } \text{ all } \quad u\in C_b(X). \end{aligned}$$

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

$$\begin{aligned} T^{\prime \prime }(C_b(X)^{\prime \prime })\subset i(E). \end{aligned}$$

Let us put

$$\begin{aligned} \widetilde{T}:=j\circ T^{\prime \prime }\circ \pi :B(\mathcal{B})\longrightarrow E \end{aligned}$$

and

Note that

$$\begin{aligned} \widetilde{T}=j\circ \hat{T} \quad \text{ and } \quad m_T=j\circ \hat{m}_T:\mathcal{B}\longrightarrow E. \end{aligned}$$

Then for each $e^{\prime }\in E^{\prime }_\xi$ we have

$$\begin{aligned} (\hat{m}_T)_{e^{\prime }}(A)=(e^{\prime }\circ m_T)(A) \quad \text{ for } \text{ each } \quad A\in \mathcal{B}. \end{aligned}$$

It follows that for each $\mathcal{A}\in \mathcal{E}_\xi$ and $A\in \mathcal{B}$ we have

$$\begin{aligned} q_\mathcal{A}(\hat{m}_T(A))=p_\mathcal{A}(m_{T}(A)). \end{aligned}$$

(2.1)

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

$$\begin{aligned} \sup _{T\in \mathcal{K}}p_\mathcal{A}(T(u_n))=\sup _{T\in \mathcal{K}}(\sup \{|e^{\prime }(T(u_n))|:e^{\prime }\in \mathcal{A}\})\rightarrow 0. \end{aligned}$$

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

$$\begin{aligned} \sup _{T\in \mathcal{K}} q_\mathcal{A}(\hat{m}_T(Z_n))=\sup _{T\in \mathcal{K}}(\sup \{|(\hat{m}_T)_{e^{\prime }}(Z_n)| :e^{\prime }\in \mathcal{A}\})\rightarrow 0. \end{aligned}$$

This means that $\hat{m}_T(Z_n)\rightarrow 0$ in $\xi _\varepsilon$ uniformly for $T\in \mathcal{K}$. Moreover, we have

$$\begin{aligned} \sup \{|(\hat{m}_T)_{e^{\prime }}(A)|:T\!\in \!\mathcal{K},e^{\prime }\in \mathcal{A},A\in \mathcal{B}\}\!\le \! \sup \{|(\hat{m}_T)_{e^{\prime }}|(X):T\in \mathcal{K},e^{\prime }\in \mathcal{A}\}<\infty . \end{aligned}$$

It follows that

$$\begin{aligned} \sup \{q_\mathcal{A}(\hat{m}_T(A)):T\in \mathcal{K},A\in \mathcal{B}\}<\infty , \end{aligned}$$

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

$$\begin{aligned}&\sup \{|(\hat{m}_T)_{e^{\prime }}|(X):T\in \mathcal{K},e^{\prime }\in \mathcal{A}\}\le 4\sup \{|(\hat{m}_T)_{e^{\prime }}(A)|:T\\&\quad \in \mathcal{K},e^{\prime }\in \mathcal{A},A\in \mathcal{B}\}<\infty ,\qquad \end{aligned}$$

and moreover, for each sequence $(Z_n)$ in $\mathcal{Z}$ such that $Z_n\downarrow \emptyset ,$ we have

$$\begin{aligned} \sup _{T\in \mathcal{K}} q_\mathcal{A}(\hat{m}_T(Z_n))\rightarrow 0, \quad \mathrm{i.e.,} \quad \sup \{|(\hat{m}_T)_{e^{\prime }}(Z_n)|:T\in \mathcal{K},e^{\prime }\in \mathcal{A}\}\rightarrow 0. \end{aligned}$$

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 $E$.

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 $E$.

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

$$\begin{aligned} \Vert m\Vert _p(A):=\sup \{|e^{\prime }\circ m|(A):e^{\prime }\in V^o_p\} \quad \text{ for } \quad A\in \mathcal{B}o, \end{aligned}$$

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

$$\begin{aligned} T(u)=\int \limits _X udm_T=\int ud\widetilde{m}_T \quad \text{ for } \text{ all } \quad u\in C_b(X). \end{aligned}$$

(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

$$\begin{aligned} (e^{\prime }\circ T)(u)=\int \limits _X u d(e^{\prime }\circ m_T)=\int \limits _X u d(\widetilde{e^{\prime }\circ m_T})= \int \limits _X u d(e^{\prime }\circ \widetilde{m}_T) \end{aligned}$$

(5.1)

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

$$\begin{aligned} e^{\prime }\circ \widetilde{m}_T=\widetilde{e^{\prime }\circ m_T} \quad \text{ for } \text{ each } \quad e^{\prime }\in E^{\prime }_\xi . \end{aligned}$$

(5.2)

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 $\sup _{T\in \mathcal{K}}\Vert \widetilde{m}_T\Vert _p(X\backslash K)\le \varepsilon )$.

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

$$\begin{aligned} \sup _{T\in \mathcal{K}}\Vert \widetilde{m}_T\Vert _p(X)=\sup \{|e^{\prime }\circ \widetilde{m}_T|(X):T\in \mathcal{K},e^{\prime }\in V^o_p\}<\infty , \end{aligned}$$

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

$$\begin{aligned} \sup \{|e^{\prime }\circ \widetilde{m}_T|(X):T\in \mathcal{K},e^{\prime }\in V^o_p\}<\infty \end{aligned}$$

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 $E$.

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

$$\begin{aligned} p_v(u)=\sup \{|u(x)v(x)|:x\in X\} \quad \text{ for } \quad u\in C_b(X), \end{aligned}$$

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

$$\begin{aligned} \int \limits _X ud\widetilde{m}=\xi -\lim \int \limits _X ud\widetilde{m}_{T_n} \end{aligned}$$

for all $u\in B(\mathcal{B}o)$. Let

$$\begin{aligned} T_{\widetilde{m}}(u)=\int \limits _X ud\widetilde{m}\quad \text{ for } \text{ all } \quad u\in B(\mathcal{B}o). \end{aligned}$$

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. $\square$