Nuclear operators on spaces of continuous vector-valued functions

Abstract

Let $\Omega$ be a compact Hausdorff space, let $E$ be a Banach space, and let $C(\Omega, E)$ stand for the Banach space of all $E$-valued continuous functions on $\Omega$ under supnorm. In this paper we study when nuclear operators on $C(\Omega, E)$ spaces can be completely characterized in terms of properties of their representing vector measures. We also show that if $F$ is a Banach space and if $T:\ C(\Omega, E)\rightarrow F$ is a nuclear operator, then $T$ induces a bounded linear operator $T^\#$ from the space $C(\Omega)$ of scalar valued continuous functions on $\Omega$ into $\slN(E,F)$ the space of nuclear operators from $E$ to $F$, in this case we show that $E^*$ has the Radon-Nikodym property if and only if $T^\#$ is nuclear whenever $T$ is nuclear.