On the compact operators case of the Bishop-Phelps-Bollob\'as property for numerical radius

TL;DR

This paper investigates the Bishop-Phelps-Bollobás property for numerical radius specifically for compact operators, establishing its presence in certain function spaces and Hilbert spaces.

Contribution

It demonstrates that $C_0(L)$ spaces, real Hilbert spaces, and isometric preduals of $\, ext{l}_1$ possess the BPBp-nu for compact operators, with new techniques for property transfer and approximation.

Findings

$C_0(L)$ spaces have BPBp-nu for compact operators.

Real Hilbert spaces have BPBp-nu for compact operators.

Preduals of $ ext{l}_1$ have BPBp-nu for compact operators.

Abstract

We study the Bishop-Phelps-Bollob\'as property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that $C_0(L)$ spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space $L$. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of $C_0(L)$ spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of $\ell_1$ have the BPBp-nu for compact operators.