Linear mappings preserving the diameter of the numerical range
type: Item
instance of: scholarly article
title: Linear mappings preserving the diameter of the numerical range
zbMATH Open document ID: 1422.47041
author: Jor-Ting Chan
author: Kong Chan
published in: Linear Algebra and its Applications
publication date: 2017-08-03T00:00:00Z
review text: Let\(\mathcal{\;H}\) be a complex Hilbert space and \(\mathcal{B(H)}\) the algebra of all bounded linear operators on \(\mathcal{H}\). For \(A\in \mathcal{B(H)}\), the numerical range of \(A\) is defined by \(W(A)=\left\{ \left\langle Ax,x\right\rangle :x\in \mathcal{H}\text{ and }\left\| x\right\| =1\right\} \). The diameter of \(W(A)\) is defined as \[ d(W(A))=\sup \left\{ \left| \lambda -\mu \right| :\lambda ,\mu \in W(A)\right\} . \] We say that a map \(\Psi :\mathcal{B(H)\rightarrow B(H)}\) preserves the diameter if \[ d(\Psi (W(A)))=d(W(A)) \] for all \(A\in \mathcal{B(H)}\). In the paper under review, authors study linear maps on \(\mathcal{B(H)}\) that preserve the diameter. They prove the following result. Theorem. Let \(\Psi :\mathcal{B(H)}\rightarrow\mathcal{B(H)}\) be a surjective linear map that preserves the diameter. Then there exist \(\lambda \in \mathbb{C}\) with \(\left| \lambda \right| =1\), a unitary operator \(U\in \mathcal{B(H)}\), and a linear functional \(\Lambda :\mathcal{B(H)}\rightarrow \mathbb{C}\) satisfying \(\Lambda (I)+\lambda \neq 0\) such that \(\Psi \) is either of the form \[ \Psi (A)=\lambda UAU^{\ast }+\Lambda (A)I\quad \text{for every }A\in \mathcal{B(H)} \] or of the form \[ \Psi (A)=\lambda UA^{t}U^{\ast }+\Lambda (A)I\quad \text{for every }A\in \mathcal{B(H)}, \] where \(I\) is the identity operator on \(\mathcal{H}\) and \(A^{t}\) denotes the transpose of \(A\) taken with respect to an orthonormal basis fixed in advance. The proof of this theorem is divided into two main parts. Let \(\left\langle I\right\rangle \) denote the linear span of \(I\). Consider the dual space \(\left( \mathcal{B(H)}/\left\langle I\right\rangle \right) ^{\ast }\) of the quotient space \(\mathcal{B(H)}/\left\langle I\right\rangle \). In the first part of the proof, authors give a characterization of the extreme functionals, i.e., the extreme points of the closed unit ball of \(\left( \mathcal{ B(H)}/\left\langle I\right\rangle \right) ^{\ast }\). T…
reviewed by: Janko Marovt
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 15A86
Mathematics Subject Classification ID: 14R10
zbMATH DE Number: 6756330
zbMATH Keywords: numerical range
zbMATH Keywords: preserver
zbMATH Keywords: diameter
MaRDI profile type: Publication
full work available at URL: j.laa.2017.05.018
OpenAlex ID: W2616773791
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DOI: 10.1016/J.LAA.2017.05.018
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description: scientific article; zbMATH DE number 6756330