The Bartle-Dunford-Schwartz and the Dinculeanu-Singer theorems revisited

Let X and Y be Banach spaces and let Ω be a compact Hausdorff space. By the classical Bartle-Dunford-Schwartz theorem, any operator S∈L(C(Ω),Y) admits an integral representation with respect to a Y⁎⁎-valued measure. By the Dinculeanu-Singer theorem, each operator U∈L(C(Ω,X),Y) admits an integral representation with respect to an L(X,Y⁎⁎)-valued measure. We establish an integral representation for any operator S∈L(C(Ω),L(X,Y)) with respect to an L(X,Y⁎⁎)-valued measure. This far-reaching extension of the Bartle-Dunford-Schwartz theorem serves as a departure point for a general integral representation theory developed in the present paper. In particular, it is an efficient tool that enables us to give an alternative simple proof to the Dinculeanu-Singer theorem. The latter theorem is proved in a more general context of X-valued continuous functions with p-compact range. Among others, useful formulas which connect different vector measures are deduced.