Spectral conditions and band reducibility of operators
type: Item
instance of: scholarly article
title: Spectral conditions and band reducibility of operators
zbMATH Open document ID: 1259.47050
author: J. Bernik
author: L. W. Marcoux
author: H. Radjavi
publication date: 2012-08-29T00:00:00Z
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 6073816
zbMATH Keywords: band triangularisability
zbMATH Keywords: band reducibility
zbMATH Keywords: positive compact operators
MaRDI profile type: Publication
full work available at URL: jdr084
OpenAlex ID: W2098986081
DOI: 10.1112/JLMS/JDR084
Recommended article: Irreducible semigroups of positive operators on Banach lattices
Recommended article: Extensions of Perron-Frobenius theory
Recommended article: Positive root vectors
Recommended article: Strong Monotonicity of Spectral Radius of Positive Operators
Recommended article: Q3800593
Recommended article: Generic existence and uniqueness of positive eigenvalues and eigenvectors
Recommended article: Q3719332
Recommended article: Q4285030
Recommended article: On the peripheral spectrum of positive operators
Recommended article: POSITIVE COMPACT OPERATORS
published in: Journal of the London Mathematical Society. Second Series
review text: Let \(M\) be a (complex) matrix with non-negative entries. Then the diagonal of \(M\) exhibits the eigenvalues of \(M\) with their multiplicity if and only if there exists a permutation matrix \(P\) such that \(P^{-1}MP\) is triangular with the same diagonal as \(M\). This pleasant fact is not very complicated to check (see, for instance, Theorem \(5.1.7\) in [\textit{H. Radjavi} and \textit{P. Rosenthal}, Simultaneous triangularization. Universitext. New York, NY: Springer (2000; Zbl 0981.15007)]). The authors of this remarkable paper try to extend this result to positive operators on \(L^{p}\)-spaces. They obtain some affirmative results under extra conditions of compactness, viz., the operator under consideration (or, at least some power of it) has to be compact. In this regard, examples are given to justify such additional hypotheses. A typical result is that, if \(T\) is a band-irreducible bounded positive operator \(T\) on \(L^{p}\left( X,\mu\right) \), then the spectral radius of any (proper) compact band compression of \(T\) is strictly smaller than that of \(T\). The reader is encouraged to read the whole paper to discover alternative (and very interesting) results.
reviewed by: Karim Boulabiar
description: scientific article; zbMATH DE number 6073816