Operators on spaces of vector-valued continuous functions

Abstract

Let K be a compact Hausdorff space and E,F B-spaces. Let L(E,F) be the space of all bounded linear operators from E into F, and C(K,E) the space of all continuous E-valued functions defined on K equipped with the sup-norm. Any bounded linear operator T:C(K,E)→F has an integral representation as Tf=∫Kfdm, where m is a measure defined on the Borel sets of K with values in L(E,F′′). One of the principal problems in this area is to characterize the various classes of operators such as the weakly compact, Dunford-Pettis, Dieudonné and unconditionally converging operators in terms of their representing measure. In this paper the author gives an expository account of the current status of these problems and indicates many open problems in this area.