A note on spectrally dominant norms
type: Item
instance of: scholarly article
title: A note on spectrally dominant norms
zbMATH Open document ID: 0983.46009
author: Ángel Rodríguez Palacios
published in: Journal of Algebra
publication date: 2002-07-15T00:00:00Z
review text: Let \(A\) be a finite-dimensional associative complex algebra. A vector space norm on it is called stable if there exists a positive constant \(\sigma\) such that \(\|x^n\|\leq\sigma\|x\|^n\) for every \(x\in A\) and all \(n\). It is known that stable norms on \(A\) are spectrally dominant (i.e., for every \(x\in A\), the spectral radius of \(x\leq\|x\|\)). A theorem of \textit{S. Friedland} and \textit{C. Zenger} [Linear Algebra Appl. 58, 97-107 (1984; Zbl 0543.15018)] asserts the converse. This theorem is not true for arbitrary \(A\), even if it is commutative and has a unit. The paper studies the limits of the theorem and shows that \(\text{Rad}(A)\) is a direct summand of \(A\) is sufficient to ensure that all spectrally dominant norms are stable. This condition becomes necessary if \(A\) is either commutative or has a unit. Also studied is a general version of the Friedland-Zenger theorem with a notion of almost stable norms.
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 1539379
zbMATH Keywords: vector space norm
zbMATH Keywords: spectrally dominant norms
zbMATH Keywords: almost stable norms
reviewed by: Srinivasa Swaminathan
MaRDI profile type: Publication
full work available at URL: jabr.2000.8390
OpenAlex ID: W2039330192
cites work: Propriétés spectrales des algèbres de Banach
cites work: Q3215824
cites work: All spectral dominant norms are stable
DOI: 10.1006/JABR.2000.8390
description: scientific article; zbMATH DE number 1539379