Characterizations of weakly compact operators on

Let be a locally compact Hausdorff space and let , is continuous and vanishes at infinity} be provided with the supremum norm. Let and be the -rings generated by the compact subsets and by the compact subsets of , respectively. The members of are called -Borel sets of since they are precisely the -bounded Borel sets of . The members of are called the Baire sets of . denotes the dual of . Let be a quasicomplete locally convex Hausdorff space. Suppose is a continuous linear operator. Using the Baire and -Borel characterizations of weakly compact sets in 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 to be weakly compact, several of which are new. The independent results on the regularity and on the regular Borel extendability of -additive -valued Baire measures are deduced as an immediate consequence of these characterizations. Some other applications are also included.