Bochner representable operators on variable exponent Lebesgue spaces

Abstract

Let $L^{p(\cdot )}(\Omega )$ be a variable exponent Lebesgue space and $X$ denote a Banach space. It is shown that a bounded linear operator $T:L^{p(\cdot )}(\Omega )\to X$ is Bochner representable if and only if $|||T||{|}_{p(\cdot )}<\infty$ (here $|||T||{|}_{p(\cdot )}$ denotes the norm of $T$ in the Dinculeanu sense). As an application, we study the compactness property of these operators.