Holomorphic maps preserving parts of the local spectrum
type: Item
instance of: scholarly article
title: Holomorphic maps preserving parts of the local spectrum
zbMATH Open document ID: 1335.47023
author: Constantin Costara
published in: Archiv der Mathematik
publication date: 2015-09-01T00:00:00Z
review text: Let \(M_n\) denote the algebra of all \(n \times n\) complex matrices. It is well known that linear maps that preserve the spectrum or the spectral radius are of a standard form. This result has been extended in several directions. For example \textit{M. González} and \textit{M. Mbekhta} [Linear Algebra Appl. 427, No. 2--3, 176--182 (2007; Zbl 1127.15005)] have considered linear maps between matrix spaces that preserve the local spectrum. More recently, \textit{A. Bourhim} and \textit{V. G. Miller} [Stud. Math. 188, No. 1, 67--75 (2008; Zbl 1145.47005)] studied linear maps that preserve the local spectral radius at a point. In the present paper, the author extends these results to the following setting: fix a nonzero \(x_{0} \in M_{n}\), let \(U\) be a domain in \(M_n\) containing the zero matrix, and let \(\varphi : U \rightarrow M_n\) be a holomorphic map such that the local spectrum of \(T\) at \(x_0\) and the local spectrum of \(\varphi(T)\) at \(x_0\) have always a point in common, for all \(T \in U\). Then \(\varphi\) preserves the spectrum, i.e., \(T\) and \(\varphi (T)\) have the same spectrum for all \(T \in U\). It is also proved that a holomorphic map \(\varphi : U \rightarrow M_{n}\) with \(\varphi (0)=0\) that preserves the local spectral radius for every element of \(U\) at \(x_0\) satisfies that there exists a modulus one complex number \(\xi\) such that \(T\) and \(\xi\varphi(T)\) have the same spectrum for all \(T\) in \(U\). The author also gives some examples about the sharpness of the results.
reviewed by: Jerónimo Alaminos Prats
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A11
zbMATH DE Number: 6477368
zbMATH Keywords: preservers
zbMATH Keywords: local spectrum
zbMATH Keywords: local spectral radius
zbMATH Keywords: holomorphic
MaRDI profile type: Publication
full work available at URL: s00013-015-0792-z
OpenAlex ID: W1002660510
cites work: Q3154380
cites work: On linear transformations preserving at least one eigenvalue
cites work: Linear maps on Mn(C) preserving the local spectral radius
cites work: On local irreducibility of the spectrum
cites work: Linear maps on \(M_{n}(\mathbb C)\) preserving the local spectrum
cites work: Q5515607
cites work: Q4951127
DOI: 10.1007/S00013-015-0792-Z
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description: scientific article; zbMATH DE number 6477368