Additive preservers of numerical range
type: Item
instance of: scholarly article
title: Additive preservers of numerical range
zbMATH Open document ID: 0998.15038
DOI: 10.1016/S0024-3795(01)00498-0
author: Gorazd Lešnjak
published in: Linear Algebra and its Applications
publication date: 2002-05-15T00:00:00Z
full work available at URL: abstract.html
review text: Let \({\mathbf C}\) be the field of complex numbers, \(M_n\) be the set of all \(n\times n\)-matrices, \(T_n\) be the set of all upper triangular \(n\times n\)-matrices with complex coefficients, \(\langle x,y \rangle\) denotes the scalar product of two vectors, \(\|x\|^2=\langle x,x\rangle\). The numerical range of a matrix \(A\in M_n\) is the set \(W(A)=\{\langle Ax,x\rangle \mid x\in {\mathbf C}^n\), \(\|x\|=1\}\). In the past two decades linear transformations on matrix algebras that preserve numerical range and various generalizations were the objects of intensive study. For the details one may visit the homepage of Chi-Kwong Li: \url{http://www.math.wm.edu/~ckli}. In this paper additive transformations preserving a numerical range on \(M_n\), \(T_n\), diagonal, block-diagonal, and block upper-triangle subalgebras of \(M_n\) are characterized. It is proved that in the case of \(M_n\) and its diagonal and block-diagonal subalgebras all such transformations are necessarily linear and exhausted by transposition and diagonal unitary similarity (similarity factors are unitary and have the same block-diagonal structure as matrices from subalgebra under consideration). For \(T_n\) and block-triangular subalgebras of \(M_n\) nonlinear additive numerical range preservers are found. The author establishes a complete classification of additive numerical range preservers over \(T_n\) and block upper-triangular subalgebras of \(M_n\). In the case of \(T_n\) the transformations under consideration are unitary similarity, transposition over the anti-diagonal, and the nonlinear transformation that maps \(a_{11}\) into itself, \(a_{1i}\) into its complex conjugate for all \(i=2,\ldots,n\), and the remaining block into its transposed over the anti-diagonal.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 1741581
zbMATH Keywords: numerical range
zbMATH Keywords: additive mapping
zbMATH Keywords: linear preserver
zbMATH Keywords: linear transformations
zbMATH Keywords: matrix algebras
zbMATH Keywords: unitary similarity
reviewed by: Alexander Guterman
MaRDI profile type: Publication
cites work: Additive rank-one preserving mappings on triangular matrix algebras
cites work: Linear operators preserving generalized numerical ranges and radii on certain triangular algebras of matrices
cites work: Numerical range. The field of values of linear operators and matrices
cites work: Q4040931
cites work: Linear Operators Preserving the Numerical Radius of Matrices
cites work: Linear operators preserving the numerical range (radius) on triangular matrices
cites work: On operators preserving the numerical range
cites work: Additive mappings preserving operators of rank one
cites work: Numerical range preserving operators on a Banach algebra
cites work: Q4515201
cites work: Linear preservers of minimal rank
Wikidata QID: Q126989403
description: scientific article; zbMATH DE number 1741581