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arXiv · 1303.6078

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

Yun Sung Choi·Sun Kwang Kim·Han Ju Lee·Miguel Martín
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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.

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Yun Sung Choi, Sun Kwang Kim, Han Ju Lee, Miguel Martín. 2013-03-25. The Bishop-Phelps-Bollob\'{a}s theorem for operators on $L_1(\mu)$. https://arxiv.org/abs/1303.6078

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