A Generalized Spectral Radius Formula and Olsen's Question

TL;DR

This paper establishes a generalized spectral radius formula for $C^*$-algebras, provides partial answers to Olsen's question on polynomial spectral norms, and explores conditions for simultaneous similarity to contractions.

Contribution

It generalizes spectral radius formulas in $C^*$-algebras, addresses Olsen's open question on polynomial spectral norms, and characterizes when commuting operators are simultaneously similar to contractions.

Findings

The formula $ ext{max}igrace r(x), orm{ x}igrace = ext{inf} orm{(1+i)^{-1}x(1+i)}$ is proven.

The infimum in the spectral radius formula is attained when $r(x)< orm{ x}$.

Operators with certain commutation and similarity properties are simultaneously similar to contractions.

Abstract

Let $A$ be a $C^*$-algebra and $I$ be a closed ideal in $A$. For $x\in A$, its image under the canonical surjection $A\to A/I$ is denoted by $\dot x$, and the spectral radius of $x$ is denoted by $r(x)$. We prove that $$\max\{r(x), \|\dot x\|\} = \inf \|(1+i)^{-1}x(1+i)\|$$ (where infimum is taken over all $i\in I$ such that $1+i$ is invertible), which generalizes spectral radius formula of Murphy and West \cite{MurphyWest} (Rota for $\mathcal{B(H)}$ \cite{Rota}). Moreover if $r(x)< \|\dot x\|$ then the infimum is attained. A similar result is proved for commuting family of elements of a $C^*$-algebra. Using this we give a partial answer to an open question of C. Olsen: if $p$ is a polynomial then for "almost every" operator $T\in B(H)$ there is a compact perturbation $T+K$ of $T$ such that $$\|p(T+K)\| = \|p(T)\|_e.$$ We show also that if operators $A,B$ commute, $A$ is similar to a contraction and $B$ is similar to a strict contraction then they are simultaneously similar to contractions.