The Bishop-Phelps-Bollob\'as property for the numerical radius: a Zizler-type approach

TL;DR

This paper studies the Bishop-Phelps-Bollobás property for the numerical radius in Banach spaces, establishing new results for specific spaces and exploring the property’s behavior in various contexts.

Contribution

It introduces a Zizler-type approach to the BPBp-nu, providing the first example of a space where numerical radius operators are dense but BPBp-nu fails.

Findings

The space ℓ_∞ satisfies the BPBp-nu.

The space ℓ_1 ⊕_∞ c_0 does not satisfy the BPBp-nu.

Strengthens previous results on the interplay between BPBp and BPBp-nu.

Abstract

We investigate the Bishop-Phelps-Bollob\'as property for the numerical radius (BPBp-nu) through a Zizler-type perspective on the classical Bishop-Phelps-Bollob\'as property (BPBp). This approach allows us to establish two new results: the real Banach space $\ell_\infty$ satisfies the BPBp-nu, while the complex space $\ell_1 \oplus_\infty c_0$ does not. Note that the latter provides the first natural example (constructed without renorming techniques) of a Banach space where the numerical radius attaining operators are dense but the BPBp-nu fails. Along the way, we strengthen the main results of the paper [Kim et al, On the Bishop-Phelps-Bollob\'as theorem for operators and numerical radius, Studia Math., 2016] concerning the interplay between the BPBp for the pair $(X,Y)$ and the BPBp-nu for a direct sum $X\oplus Y$ of Banach spaces. We further explore the validity of the Zizler-type BPBp across different pairs of Banach spaces, and how this property relates to the classical BPBp and the BPBp-nu. Finally, we specialize our analysis to the framework of compact operators.