Integral Representation of Continuous Operators with Respect to Strict Topologies - Nowak, Marian (Resultate der Mathematik, 2017)
Let X be a completely regular Hausdorff space and B o be the σ -algebra of Borel sets in X . Let C b ( X ) (resp. B ( B o ) ) be the space of all bounded continuous (resp. bounded B o -measurable) scalar functions on X , equipped with the natural strict topology β . We develop a general integral representation theory of ( β , ξ ) -continuous operators from C b ( X ) to a lcHs ( E , ξ ) with respect to the representing Borel measure taking values in the bidual E ξ ′ ′ of ( E , ξ ) . It is shown that every ( β , ξ ) -continuous operator T : C b ( X ) → E possesses a ( β , ξ E ) -continuous extension T ^ : B ( B o ) → E ξ ′ ′ , where ξ E stands for the natural topology on E ξ ′ ′ . If, in particular, X is a k -space and ( E , ξ ) is quasicomplete, we present equivalent conditions for a ( β , ξ ) -continuous operator T : C b ( X ) → E to be weakly compact. As an application, we have shown that if X is a k -space and a quasicomplete lcHs ( E , ξ ) contains no isomorphic copy of c 0 , then every ( β , ξ ) -continuous operator T : C b ( X ) → E is weakly compact.