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)\rightarrow 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.