Characterization of some classes of operators on spaces of vector-valued continuous functions

Abstract

LetKbe a compact Hausdorff space andE, FBanach spaces. We denote byC(K, E)the Banach space of all continuous.E-valued functions defined onK,with the supremum norm. It is well known ([6], [7]) that every operator (= bounded linear operator)TfromC(K, E)toFhas a finitely additive representing measuremof bounded semi-variation, defined on the Borel σ-field Σ ofKand with values inL(E, F″)(the space of all operators fromEinto the second dual ofF),in such a way thatwhere the integral is considered in Dinculeanu's sense.