Multipliers of certain vector valued function spaces

Abstract

Throughout this paper let G be a locally compact Hausdorff-Abelian group with Haar measure 1. Let A be a commutative Banach algebra with identity of norm one. Let Y be an essential Banach A-module and let Y* and Y** denote the dual space and second dual space of Y, respectively. We say that Y (Y* or Y**) has the wide Radon-Nikodym property with respect to p if for every measurable set E in G with p(E) < x8, Y ( Y* or Y**) has the Radon-Nikodym property with respect to pE, where pE is defined on all measurable sets F in G with Fc E by pE(F) = p(F). Let (L,(G, A), L,(G, Y)) denote the space of all multipliers from L,(G, A) to L,(G, Y). We prove in Theorem 9 that if Y has the wide Radon-Nikodym property, then (L,(G, A), .E.,( G, Y)) is isometrically isomorphic with L,(G, Y) for 1 < p < T/J.