Matrix groups with positive spectra
type: Item
instance of: scholarly article
title: Matrix groups with positive spectra
zbMATH Open document ID: 0779.47035
DOI: 10.1016/0024-3795(92)90422-7
author: Aleksander Simonič
published in: Linear Algebra and its Applications
publication date: 1992-09-27T00:00:00Z
review text: A group \({\mathcal G}\) of operators acting on a real or complex vector space \({\mathcal V}\) is called divisible if for every \(A\in{\mathcal G}\) there exists an operator \(B\in{\mathcal G}\) such that \(B^ 2=A\). Now let \({\mathcal G}\) be a divisible group of operators with positive spectra acting on the vector space \({\mathcal V}\) such that \({\mathcal G}\) is relatively closed in the group \({\mathcal L}({\mathcal V})\) of all invertible operators on \({\mathcal V}\). Then there is a bijection \(\Phi:{\mathcal G}\to{\mathcal L}=\log({\mathcal G})\), and it is shown that \({\mathcal L}\) is a real exponentially invariant Lie algebra (i.e. \(e^ ABe^{-A}\in{\mathcal L}\) for all \(A,B\in{\mathcal L})\) whose elements have only real spectra. Since the commutators of \({\mathcal L}\) are nilpotent the Lie algebra \({\mathcal L}\) turns out to be triangularizable. Thus also \({\mathcal G}\) itself is triangularizable. Furthermore, the author gives a construction of irreducible matrix groups whose elements have positive trace. Examples show that the divisibility condition on \({\mathcal G}\) is necessary, and that the results on the triangularizability do not apply to semigroups of operators.
reviewed by: Huberta Lausch
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20G20
zbMATH DE Number: 64137
zbMATH Keywords: divisible group of operators with positive spectra
zbMATH Keywords: Lie algebra
zbMATH Keywords: triangularizable
zbMATH Keywords: irreducible matrix groups
zbMATH Keywords: positive trace
zbMATH Keywords: semigroups of operators
MaRDI profile type: Publication
full work available at URL: 0024-3795(92)90422-7
OpenAlex ID: W2071281159
cites work: Totally positive matrices
cites work: Q3254327
cites work: Q4138841
cites work: Q4118027
cites work: The Engel-Jacobson theorem revisited
description: scientific article; zbMATH DE number 64137