Linear maps relating different unitary similarity orbits or different generalized numerical ranges
type: Item
instance of: scholarly article
title: Linear maps relating different unitary similarity orbits or different generalized numerical ranges
zbMATH Open document ID: 0833.15024
DOI: 10.1016/0024-3795(95)00105-Z
author: Chi-Kwong Li
author: Nam-Kiu Tsing
published in: Linear Algebra and its Applications
publication date: 1995-09-06T00:00:00Z
review text: Let \(\mathcal M\) denote either the complex linear space \(M_n\) of \(n \times n\) complex matrices or the real linear space \(H_n\) of \(n \times n\) Hermitian matrices. The unitary similarity orbit of \(C \in {\mathcal M}\) is defined by \({\mathcal U} (C) = \{UCU^* : U \text{ unitary}\}\) and the corresponding circular unitary orbit is the set \({\mathcal V} (C) = \{\mu X : \mu\) a scalar, \(|\mu|= 1\), \(X \in {\mathcal U} (C)\}\). For \(A \in {\mathcal M}\) define the \(C\)-numerical range and the \(C\)-numerical radius by \(W_c (A) = \{\text{tr} (AX) : X \in {\mathcal U} (C)\}\) and \(r_c = \max \{|z|: z \in W_c(A)\},\) respectively. Let \(C, D \in H_n\). The authors study linear operators \(T\) on \(\mathcal M\) satisfying one of the following properties: (I) \(W_D(T(A)) = W_C (A)\) for all \(A \in {\mathcal M}\), (II) \(r_D (T(A)) = r_C(A)\) for all \(A \in {\mathcal M}\), (III) \({T({\mathcal U} (D)) = {\mathcal U} (C)}\), (IV) \(T({\mathcal V} (D)) = {\mathcal V} (C)\). In particular, they determine the conditions on \(C\) and \(D\) for the existence of a linear operator \(T\) satisfying any one of the conditions (I)-(IV), and characterize such an operator if it exists.
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 778270
zbMATH Keywords: unitary similarity orbit
zbMATH Keywords: circular unitary orbit
zbMATH Keywords: \(C\)-numerical range
zbMATH Keywords: \(C\)-numerical radius
reviewed by: Ali-Akbar Jafarian
MaRDI profile type: Publication
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full work available at URL: 0024-3795(95)00105-z
OpenAlex ID: W1968368108
description: scientific article; zbMATH DE number 778270