A Generalized Spectral Radius Formula and Olsen's Question (1007.4655v2)

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.