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

TL;DR

This work extends classical M-ideal theory for compact operators to spaces endowed with a numerical-radius–based norm, focusing on when $ ext{K}_v(X)$ forms a proper M-ideal in $ ext{L}_v(X)$. It provides a Werner-type characterization in the numerical-radius setting, proves a HWW-style shrinking-compact-approximation framework, and analyzes geometric properties such as proximinality and farthest points, along with a compact-perturbation property analog. The results connect the numerical index $n(X)$ and asymptotic moduli to the existence and structure of M-ideals, yielding both positive results for standard spaces (e.g., $oldsymbol{\ell_p}$ with $1<p<ty$ and $n(X)>0$) and limitations via CAP/Asplundness considerations. Collectively, the paper maps the boundary between when compact operators endowed with the numerical radius norm form M-ideals and how this affects radius attainments and perturbations, enriching the operator-space geometry under alternative norms.

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.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 2.1
• Example 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Corollary 2.5
• Corollary 2.6