$p$-variations of vector measures with respect to vector measures and integral representation of operators

Abstract

In this paper we provide two representation theorems for two relevant classes of operators from any $p$-convex order continuous Banach lattice with weak unit into a Banach space: the class of continuous operators and the class of cone absolutely summing operators. We prove that they can be characterized as spaces of vector measures with finite $p$-semivariation and $p$-variation, respectively, with respect to a fixed vector measure. We give in this way a technique for representing operators as integrals with respect to vector measures.