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

Abstract

Let T be a locally compact Hausdorff space and let Co(T) ={f: T -* C, f is continuous and vanishes at infinity} be provided with the supremum norm. Let l3B(T) and 13o(T) be the a-rings generated by the compact subsets and by the compact G6 subsets of T, respectively. The members of Bc (T) are called o-Borel sets of T since they are precisely the or-bounded Borel sets of T. The members of 130(T) are called the Baire sets of T. M(T) denotes the dual of Co(T). Let X be a quasicomplete locally convex Hausdorff space. Suppose u: Co(T) -* X is a continuous linear operator. Using the Baire and a-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 a-additive X-valued Baire measures are deduced as an immediate consequence of these characterizations. Some other applications are also included.