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 ν 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 L¹ (G,X), of strongly measurable X-valued function that are Bochner integrable functions defined on G, and other spaces of vector-valued functions.

Keywords: Banach spaces; convolution operators; weakly compact operators; Dunford-Pettis property

Quaestiones Mathematicae 32(2009), 569–585