广义谱半径公式与Olsen问题
A Generalized Spectral Radius Formula and Olsen's Question
摘要：
设$A$是$C^*$ -代数，$I$是$A$中的闭理想.对于$x\in A$，其在正则满射$A\to A/I$下的像用$\dot x$表示，并且$x$的谱半径用$r(x)$表示。本文证明了$$\max\{r(x), \|\dot x\|\} = \inf \|(1+i)^{-1}x(1+i)\|$$（其中下确界取于所有$i\in I$上使得$1+i$可逆），推广了Murphy和West \cite{MurphyWest}的谱半径公式（Rota for $\mathcal{B(H)}$ \cite{Rota}）.此外，如果$r(x)< \|\dot x\|$，则达到下确界。对$C^*$ -代数的交换元素族也证明了类似的结果.利用这一点，我们对C.奥尔森：若$p$是多项式，则对“几乎每一个”算子$T\in B(H)$，存在$T$紧扰动$T+K$，使得$$\|p(T+K)\| = \|p(T)\|_e.$$我们还证明了，若算子$A,B$可换，$A$类似于压缩，$B$类似于严格压缩，则它们同时类似于压缩。
英文摘要：
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.
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