The Bishop-Phelps-Bollob\'{a}s theorem for operators on $L_1(\mu)$

TL;DR

This paper extends the Bishop-Phelps-Bollobás theorem to various classes of operators on $L_1$ spaces and related function spaces, demonstrating its broad applicability in functional analysis.

Contribution

It proves the Bishop-Phelps-Bollobás theorem for operators between $L_1$ spaces, from $L_1$ to $L_$, and from $L_1$ to $C(K)$, covering new classes of operators.

Findings

The theorem holds for $$ to $$ operators for all measures.

It also holds for $$ to $L_$ operators with arbitrary measures.

The theorem applies to certain operators from $L_1()$ to $C(K)$, including Bochner representable and weakly compact operators.

Abstract

In this paper we show that the Bishop-Phelps-Bollob\'as 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-Bollob\'as 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.