Numerical radius and a notion of smoothness in the space of bounded linear operators

TL;DR

This paper explores how the numerical radius induces a smoothness structure in the space of bounded linear operators on certain Banach spaces, characterizing orthogonality and illustrating geometric differences from the operator norm.

Contribution

It identifies classes of Banach spaces where the numerical radius defines a norm and characterizes Birkhoff-James orthogonality in this context.

Findings

Numerical radius defines a norm on bounded linear operators for certain Banach spaces.

Characterization of Birkhoff-James orthogonality with the numerical radius norm.

Examples illustrating geometric differences between numerical radius and operator norm.

Abstract

We observe that the classical notion of numerical radius gives rise to a notion of smoothness in the space of bounded linear operators on certain Banach spaces, whenever the numerical radius is a norm. We demonstrate an important class of \emph{real} Banach space $ \mathbb{X} $ for which the numerical radius defines a norm on $ \mathbb{L}(\mathbb{X}), $ the space of all bounded linear operators on $ \mathbb{X} $. We characterize Birkhoff-James orthogonality in the space of bounded linear operators on a finite-dimensional Banach space, endowed with the numerical radius norm. Some examples are also discussed to illustrate the geometric differences between the numerical radius norm and the usual operator norm, from the viewpoint of operator smoothness.