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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