The Bishop-Phelps-Bollobás theorem for operators on $L_1(\mu )$

Abstract

In this paper we show that the Bishop-Phelps-Bollobás theorem holds for $\mathcal{L}(L_1(\mu ),L_1(\nu ))$ for all measures $\mu$ and $\nu$ and also holds for $\mathcal{L}(L_1(\mu ),L_{\infty }(\nu ))$ for every arbitrary measure $\mu$ and every localizable measure $\nu$. Finally, we show that the Bishop-Phelps-Bollobás theorem holds for two classes of bounded linear operators from a real $L_1(\mu )$ into a real $C(K)$ if $\mu$ is a finite measure and $K$ is a compact Hausdorff space. In particular, one of the classes includes all Bochner representable operators and all weakly compact operators.