A Generalized Spectral Radius Formula and Olsen's Question

Let $A$ be a $C^{\ast }$-algebra and $I$ be a closed ideal in $A$. For $x\in A$, its image under the canonical surjection $A\to A\mathrm{/}I$ is denoted by $\dot{x}$, and the spectral radius of $x$ is denoted by $r(x)$. We prove that $$\max \{r(x),\mathrm{\parallel }\dot{x}\mathrm{\parallel }\}=\inf \mathrm{\parallel }(1+i{)}^{-1}x(1+i)\mathrm{\parallel }$$ (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 $\mathcal{B}(\mathcal{H})$ \cite{Rota}). Moreover if $r(x)<\mathrm{\parallel }\dot{x}\mathrm{\parallel }$ then the infimum is attained. A similar result is proved for commuting family of elements of a $C^{\ast }$-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 $$\mathrm{\parallel }p(T+K)\mathrm{\parallel }=\mathrm{\parallel }p(T){\mathrm{\parallel }}_e\mathrm{.}$$ 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.