Best Approximation in Numerical Radius

TL;DR

This paper establishes a necessary and sufficient condition for the best approximation of compact operators in a reflexive Banach space with respect to the numerical radius, extending classical norm approximation results.

Contribution

It provides a new characterization for best approximation in numerical radius for compact operators, generalizing existing norm approximation results to this setting.

Findings

Characterization of best approximation in numerical radius

Extension of norm approximation results to numerical radius

Application to minimal extensions in operator theory

Abstract

Let $X$ be a reflexive Banach space. In this paper we give a necessary and sufficient condition for an operator $T\in \mathcal{K}(X)$ to have the best approximation in numerical radius from the convex subset $\mathcal{U} \subset \mathcal{K}(X),$ where $\mathcal{K}(X)$ denotes the set of all linear, compact operators from $X$ into $X.$ We will also present an application to minimal extensions with respect to the numerical radius. In particular some results on best approximation in norm will be generalized to the case of the numerical radius.