Complete spectral sets and numerical range

TL;DR

This paper introduces the complete numerical radius norm for operator algebra homomorphisms, relates it to the completely bounded norm, and explores implications for spectral and numerical radius sets, including a universal bound related to Crouzeix's theorem.

Contribution

It defines the complete numerical radius norm, establishes its explicit relation to the completely bounded norm, and applies this to spectral set theory and Crouzeix's theorem with new bounds.

Findings

Explicit formula for the complete numerical radius norm in terms of the completely bounded norm

Complete $C$-spectral sets imply complete $M$-numerical radius sets with $M=rac12(C+C^{-1})$

Universal constant $M<5.6$ bounds the numerical radius of polynomial functions of operators

Abstract

We define the complete numerical radius norm for homomorphisms from any operator algebra into ${\mathcal B}({\mathcal H})$, and show that this norm can be computed explicitly in terms of the completely bounded norm. This is used to show that if $K$ is a complete $C$-spectral set for an operator $T$, then it is a complete $M$-numerical radius set, where $M=\frac12(C+C^{-1})$. In particular, in view of Crouzeix's theorem, there is a universal constant $M$ (less than 5.6) so that if $P$ is a matrix polynomial and $T \in {\mathcal B}({\mathcal H})$, then $w(P(T)) \le M \|P\|_{W(T)}$. When $W(T) = \overline{\mathbb D}$, we have $M = \frac54$.