Extension of continuous linear operators on C_b(X, E)

Abstract

Let X be a completely regular Hausdorff space and E be a Banach space. Let C_b(X, E) (resp. L^∞(Bo, E)) be the space of all bounded continuous (resp. bounded strongly Borel-measurable) functions f : X → E, equipped with the natural mixed topologies. Then C_b(X, E) ʗL^∞(Bo, E) whenever X or E is separable. We study the problem of extension of different classes of continuous linear operators acting from C_b(X, E) to a Banach space F to corresponding continuous linear operators acting from L^∞(Bo, E) to F.