Spectral sets and operator radii

TL;DR

This paper investigates various operator radii for homomorphisms from operator algebras to bounded operators, providing explicit formulas and applications to spectral sets and numerical radius concepts.

Contribution

It introduces explicit computations of operator radii in terms of the norm and connects spectral sets with numerical radius sets, extending previous results.

Findings

Operator radii can be explicitly computed using the usual norm.

Spectral sets are shown to be numerical radius sets with a specific bound.

General results for operator radii related to $ ho$-dilations are established.

Abstract

We study different operator radii of homomorphisms from an operator algebra into $B(H)$ and show that these can be computed explicitly in terms of the usual norm. As an application, we show that if $\Omega$ is a $K$-spectral set for a Hilbert space operator, then it is a $M$-numerical radius set, where $M=\frac{1}{2}(K+K^{-1})$. This is a counterpart of a recent result of Davidson, Paulsen and Woerdeman. More general results for operator radii associated with the class of operators having $\rho$-dilations in the sense of Sz.-Nagy and Foias are given. A version of a result of Drury concerning the joint numerical radius of non-commuting $n$-tuples of operators is also obtained.