The Bishop–Phelps–Bollobás theorem for operators

Abstract

We prove the Bishop–Phelps–Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop–Phelps–Bollobás theorem holds for operators from ℓ 1 into Y. Several examples of classes of such spaces are provided. For instance, the Bishop–Phelps–Bollobás theorem holds when the range space is finite-dimensional, an L 1 ( μ ) -space for a σ-finite measure μ, a C ( K ) -space for a compact Hausdorff space K, or a uniformly convex Banach space.