Journal of Operator Theory

Volume 54, Issue 2, Fall 2005 pp. 291-303.

Characterizing isomorphisms between standard operator algebras by spectral functions

Authors: Zhaofang Bai (1) and Jinchuan Hou (2)
Author institution: (1) School of Science, Xian Jiaotong University, Xian, 710049, P. R. China; Department of Mathematics, Shanxi Teachers University, Linfen, 041004, P. R. China, (2) Department of Mathematics, Shanxi Teachers University, Linfen, 041004, P. R. China; Department of Mathematics, Shanxi University, Taiyuan, 030000, P. R. China

Summary: Let ${\mathcal A}$ and ${\mathcal B}$ be standard operator algebras on an infinite dimensional complex Banach space $X$, and let $\Phi$ be a map from ${\mathcal A}$ onto ${\mathcal B}$. We introduce thirteen parts of spectrum for elements in ${\mathcal A}$ and ${\mathcal B}$ and let $\triangle^{\mathcal A}(T)$ denote any one of these thirteen parts of the spectrum of $T$ in ${\mathcal A}$. We show that if $\Phi$ satisfies that $\triangle^{\mathcal A}(T+ S)=\triangle ^{\mathcal B}(\Phi(T)+\Phi(S))$ and $\triangle^{\mathcal A}(T+2 S)=\triangle^{\mathcal B}(\Phi(T)+2\Phi(S))$ for all $T$, $S\in{\mathcal A}$, then $\Phi$ is either an isomorphism or an anti-isomorphism.

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