Geometry of the space of compact operators endowed with the numerical radius norm

TL;DR

This paper explores the geometric structure of the space of compact operators on Banach spaces when equipped with the numerical radius norm, revealing conditions under which it forms an M-ideal and analyzing related geometric properties.

Contribution

It characterizes when the space of compact operators with the numerical radius norm is an M-ideal, based on the numerical index of the underlying Banach space, and studies related geometric properties.

Findings

Compact operators on $ ext{ell}_p$ with non-zero numerical index form M-ideals.

Spaces containing $ ext{ell}_1$ with numerical index > 1/2 are not M-ideals.

Analysis of proximinality and farthest points in the numerical radius context.

Abstract

We investigate the space of bounded linear operators on a Banach space equipped with a norm which is equivalent to the operator norm such that the subspace of compact operators is an M-ideal. In particular, we observe that the space of compact operators on $\ell_p$ $(1<p<\infty)$ equipped with the numerical radius norm is an M-ideal whenever the numerical index of $\ell_p$ is not $0$. On the other hand, we show that the space of compact operators on a Banach space containing an isomorphic copy of $\ell_1$ whose numerical index is greater than $1/2$ is not M-ideals. We also study the proximinality, the existence of farthest points and the compact perturbation property for the numerical radius.