The Bishop-Phelps-Bollob\'as property for numerical radius in $\ell_1(\mathbb{C})$

TL;DR

This paper establishes a Bishop-Phelps-Bollobás type theorem for the numerical radius of bounded linear operators on certain complex Banach spaces, providing constructive versions for ll_1(\u00a3) and c_0(ll_1()).

Contribution

It extends the Bishop-Phelps-Bollobás theorem to the numerical radius for operators on ll_1() and c_0(), including constructive proofs.

Findings

The set of bounded linear operators on ll_1() admits a Bishop-Phelps-Bollobe1s theorem for numerical radius.

Constructive versions of the theorem are provided for ll_1().

The results do not necessarily extend to all Banach spaces.

Abstract

We show that the set of bounded linear operators from $X$ to $X$ admits a Bishop-Phelps-Bollob\'as type theorem for numerical radius whenever $X$ is $\ell_1(\mathbb{C})$ or $c_0(\mathbb{C})$. As an essential tool we provide two constructive versions of the classical Bishop-Phelps-Bollob\'as theorem for $\ell_1(\mathbb{C})$.