The Bishop-Phelps-Bollobas theorem for operators from L(1)(mu) to Banach spaces with the Radon-Nikodym property

Abstract

Let Y be a Banach space and (Omega, Sigma, mu) be a a-finite measure space, where Sigma is an infinite alpha sigma-algebra of measurable subsets of Omega. We show that if the couple (L-1 (mu), Y) has the Bishop-Phelps-Bollobas property for operators, then Y has the AHSP. Further, for a Banach space Y with the Radon-Nikodym property, we prove that the couple (L-1(mu), Y) has the Bishop-Phelps-Bollobas property for operators if and only if Y has the AHSP. (C) 2011 Elsevier Inc. All rights reserved.