On Additive Operators

Abstract

Representation theorems for additive functional have been obtained in [ 2, 4; 6-8; 10-13 ]. Our aim in this paper is to study the representation of additive operators. Let S be a compact Hausdorff space and let C ( S ) be the space of real-valued continuous functions defined on S . Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C ( S ) into X . We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” { μ h } which take their values in X** . If X is weakly sequentially complete, then { μ h } can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X * is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [ 9; 11 ; §4]):