The Bishop-Phelps-Bolloba's theorem for operators

摘要

We prove the Bishop-Phelps-Bollobas theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property P of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop-Phelps-Bollobas theorem holds for operators from l(1) into Y. Several examples of classes of such spaces are provided. For instance, the Bishop-Phelps-Bollobas theorem holds when the range space is finite-dimensional, an L-1 (mu)-space for a sigma-finite measure mu, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space. (c) 2008 Elsevier Inc. All rights reserved.