Nonlinear preservers of pseudospectral radius on Banach spaces
type: Item
instance of: scholarly article
title: Nonlinear preservers of pseudospectral radius on Banach spaces
zbMATH Open document ID: 1485.47061
author: Constantin Costara
published in: Journal of Mathematical Analysis and Applications
publication date: 2019-11-28T00:00:00Z
review text: Let \({\mathcal L}(X)\) be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space \(X\). For \(\epsilon>0\), the \(\epsilon\)--pseudospectrum of an operator \(T\in{\mathcal L}(X)\) is \(\sigma_\epsilon(T)=\left\{\lambda\in\mathbb C:\|(\lambda-T)^{-1}\|\geq\frac{1}{\epsilon}\right\},\) with the convention that \(\|(\lambda-T)^{-1}\|=\infty\) if \(\lambda-T\) is not invertible. This set is compact and the \(\epsilon\)-pseudospectral radius of \(T\) is \(r_\epsilon(T):=\max\big\{|\lambda|:\lambda\in\sigma_\epsilon(T)\big\}.\) The author proves that a surjective map \(\varphi\) on \({\mathcal L}(X)\) satisfies \[ r_\epsilon(\varphi(T)\varphi(S))=r_\epsilon(TS),\ T,S\in{\mathcal L}(X), \] if and only if there exist a complex-valued functional \(\xi\) on \({\mathcal L}(X)\) and a bijective linear or conjugate linear isometry \(U\) on \(X\) such that \(|\xi(T)|=1\) and \(\varphi(T)=\xi(T)UTU^{-1}\) for all \(T\in{\mathcal L}(X)\). When \(X=\mathbb{C}^n\) is a finite-dimensional space with any norm and \(n\geq3\), he obtains a similar result but without the surjectivity assumption on \(\varphi\). The proof of these results uses several auxiliary lemmas, some of them are new. Among the new ones, the author proves that an operator \(T\in{\mathcal L}(X)\) is zero if and only if \(r_\epsilon(T)=\epsilon\).
reviewed by: Abdellatif Bourhim
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 7138410
zbMATH Keywords: pseudospectrum
zbMATH Keywords: pseudospectral radius
zbMATH Keywords: nonlinear preservers
describes a project that uses: Eigtool
MaRDI profile type: Publication
full work available at URL: j.jmaa.2019.123562
OpenAlex ID: W2979091851
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DOI: 10.1016/J.JMAA.2019.123562
description: scientific article; zbMATH DE number 7138410