Convolution Operators on Spaces of Vector-Valued Functions that are Completely Continuous

Abstract

Let X be a Banach space and let G be a compact abelian group. We study when convolution operators that are induced by a regular Borel scalar measure v on G are Completely Continuous (or Dunford-Pettis) operators (respectively, Weak-Dunford-Pettis) when they act on C(G,X), the space of continuous X-valued functions defined on G, on the space L ¹(G,X), of strongly measurable X-valued function that are Bochner integrable functions defined on G, and other spaces of vector-valued functions.