Additive spectrum compressors
type: Item
instance of: scholarly article
title: Additive spectrum compressors
zbMATH Open document ID: 1067.47049
author: Bojan Kuzma
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-04-18T00:00:00Z
review text: Let \(X\) be a Banach space and denote with \(B(X)\) and \(F(X)\) the algebra of all bounded linear operators on \(X\) and the ideal of all of its finite rank elements, respectively. By a unital standard operator algebra on \(X\), the author means a closed unital subalgebra of \(B(X)\) which contains \(F(X)\). The aim of this paper is to describe the general form of surjective additive maps between unital standard operator algebras which compress the spectrum. The main result (Theorem 7) solves a somewhat more complicated problem and it reads as follows. Let \(X\), \(Y\) be complex Banach spaces and let \(\mathcal A\), \(\mathcal B\) be unital standard operator algebras on \(X\) and \(Y\), respectively. Suppose that \(\dim\mathcal A=\infty\). Let \(\Phi:\mathcal A\to\mathcal B\) be a surjective additive map which does not annihilate all rank-one idempotents. Suppose that for each \(S\in \mathcal A\) we have either \[ \text{Sp}_{\mathcal B}(\Phi(S))\subset \text{Sp}_{\mathcal A}(S) \] or \[ \text{Sp}_{\mathcal B}(\Phi(S))\subset\overline{\text{Sp}_{\mathcal A}(S)}, \] where \(\text{Sp}\) denotes spectrum and bar stands for complex conjugate. Then \(\Phi\) takes one of the following two standard forms: \[ \Phi(S)=ASA^{-1} \quad (S\in \mathcal A), \] \[ \Phi(S)=BS'B^{-1} \quad (S\in\mathcal A), \] where \(A:X\to Y\) and \(B:X^*\to Y\) are bounded (conjugate) linear bijections, respectively. In the latter case, \(X\) and \(Y\) are reflexive. The proof uses some important facts from the theory of analytic functions. An application of the result above to so-called local approximate (anti)multiplications is also presented.
reviewed by: Lajos Molnár
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 2157550
zbMATH Keywords: additive preserver
zbMATH Keywords: spectrum
zbMATH Keywords: standard operator algebra
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.09.004
OpenAlex ID: W2011761948
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DOI: 10.1016/J.JMAA.2004.09.004
description: scientific article; zbMATH DE number 2157550