Characterizations of weakly compact operators on $C_o(T)$

Abstract:

Let $T$ be a locally compact Hausdorff space and let $C_o(T)= \{f : T \rightarrow \mathbb {C}$, $f$ is continuous and vanishes at infinity} be provided with the supremum norm. Let $\mathcal {B}_c(T)$ and $\mathcal {B}_o(T)$ be the $\sigma$-rings generated by the compact subsets and by the compact $G_\delta$ subsets of $T$, respectively. The members of $\mathcal {B}_c(T)$ are called $\sigma$-Borel sets of $T$ since they are precisely the $\sigma$-bounded Borel sets of $T$. The members of $\mathcal {B}_o(T)$ are called the Baire sets of $T$. $M(T)$ denotes the dual of $C_o(T)$. Let $X$ be a quasicomplete locally convex Hausdorff space. Suppose $u: C_o(T) \rightarrow X$ is a continuous linear operator. Using the Baire and $\sigma$-Borel characterizations of weakly compact sets in $M(T)$ as given in a previous paper of the author’s and combining the integration technique of Bartle, Dunford and Schwartz, we obtain 35 characterizations for the operator $u$ to be weakly compact, several of which are new. The independent results on the regularity and on the regular Borel extendability of $\sigma$-additive $X$-valued Baire measures are deduced as an immediate consequence of these characterizations. Some other applications are also included.