Nonlinear maps preserving numerical radius of indefinite skew products of operators
type: Item
instance of: scholarly article
title: Nonlinear maps preserving numerical radius of indefinite skew products of operators
zbMATH Open document ID: 1172.47026
author: Kan He
author: Jinchuan Hou
author: Xiu-Ling Zhang
published in: Linear Algebra and its Applications
publication date: 2009-04-29T00:00:00Z
review text: Let \(H\) be a complex Hilbert space with \(\dim H \geq 3\) and let \(B(H)\) be the algebra of all bounded linear operators. For \(A \in B(H)\), let \(w(A)\) denote its numerical radius. Given an invertible selfadjoint operator \(J\), the indefinite conjugate of \(A \in B(H)\) with respect to \(J\) is \(A^\dagger= J^{-1}A^* J\). Let \({\mathcal V}, {\mathcal W}\) be subsets of \(B(H)\) containing all rank one operators. In the paper under review, the authors give a complete characterization of surjective maps \(\varphi: {\mathcal W} \to {\mathcal V}\) with the property that \( w (A^\dagger B)= w(\varphi (A)^\dagger \varphi ( B) )\) and \(w(A B^\dagger)= w(\varphi (A) \varphi (B)^\dagger)\) for all \(A, B \in {\mathcal W}\). They also describe maps \(\varphi: {\mathcal W} \to {\mathcal V} \) satisfying \( w( A B^\dagger A)= w( \varphi (A) \varphi (B)^\dagger \varphi (A))\) for all \(A, B \in {\mathcal W}\). Let \(\mathcal A\) be a standard algebra on \(H\). Recall that surjective maps \(\varphi: \mathcal A \to \mathcal A\) preserving the numerical radius of operator products (\(w(AB)= w( \varphi (A) \varphi (B)\) for all \(A, B \in \mathcal A\)) were studied by \textit{J.--L.\thinspace Cui} and \textit{J.--C.\thinspace Hou} [Linear Algebra Appl.\ 428, No.\,7, 1649--1663 (2008; Zbl 1146.47023)]. Now, let \(B_s (H)\) denote the subspace of all selfadjoint operators in \(B(H)\). It should be pointed out that surjective maps \(\varphi: B_s (H) \to B_s (H)\) satisfying \(w(AB)= w( \varphi (A) \varphi (B))\) (respectively, \(w(ABA)= w( \varphi (A) \varphi (B) \varphi (A))\)) for all \(A, B \in B_s (H)\) have been previously determined by the authors in a preprint.
reviewed by: Nadia Boudi
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 47H20
zbMATH DE Number: 5549278
zbMATH Keywords: nonlinear map
zbMATH Keywords: numerical radius
zbMATH Keywords: indefinite skew product of operators
MaRDI profile type: Publication
full work available at URL: j.laa.2008.12.002
OpenAlex ID: W2053073159
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DOI: 10.1016/J.LAA.2008.12.002
description: scientific article; zbMATH DE number 5549278