On the Bishop-Phelps-Bollobás property for numerical radius

Abstract

We study the Bishop-Phelps-Bollobás property for numerical radius (in short, BPBp-nu) and find sufficient conditions for Banach spaces ensuring the BPBp-nu. Among other results, we show that $L_1(\mu )$-spaces have this property for every measure $\mu$. On the other hand, we show that every infinite-dimensional separable Banach space can be renormed to fail the BPBp-nu. In particular, this shows that the Radon-Nikodým property (even reflexivity) is not enough to get BPBp-nu.

References in corpus (1)

• Recent progress and open questions on the numerical index of Banach spaces

Cited by in corpus (2)

• The Bishop-Phelps-Bollob{á}s property for numerical radius of operators on $L_1(\mu )$
• On the Crawford number attaining operators