Triangular form for bounded linear operators
type: Item
instance of: scholarly article
title: Triangular form for bounded linear operators
zbMATH Open document ID: 0709.47020
DOI: 10.1016/0022-1236(90)90122-2
author: Robert R. Rogers
published in: Journal of Functional Analysis
publication date: 1990-01-01T00:00:00Z
review text: The canonical Jordan form for matrices has been extended in many ways to bounded operators on infinite dimensional Hilbert spaces. The spectral operators T of N. Dunford constitute a generic example from that point of view: \(T=S+Q\), where S is ``diagonalizable'' by a spectral measure, Q is quasi-nilpotent and \([S,Q]=0\), see \textit{N. Dunford}, and \textit{J. T. Schwartz}, Linear operators. Part III, New York (1971; Zbl 0243.47001). The aim of the paper under review is to prove the existence of a similar decomposition for operators T with sufficiently many invariant subspaces which divide the spectrum of T. Quite specifically, if there exists a finitely additive spectral measure of T with respect to a Boolean algebra of subsets of \({\mathbb{C}}\) which contains arbitrary small sets (in the sense of the diameter), then \(T=N+Q\), where N is a normal operator and Q is quasi-nilpotent. Moreover, it is shown that \(\sigma (N)=\sigma (T)\) and there exists a nest of invariant subspaces of N which separates the points of \(\sigma\) (T). The approach is similar but not reducible to the classical theory of Volterra operators, in the sense of Gohberg-Krein and Brodskij. A similar decomposition is proved to hold for Riesz elements in \(C^*\)-algebras modulo closed ideals. The paper ends with two interesting open problems.
Mathematics Subject Classification ID: 47A66
Mathematics Subject Classification ID: 47B40
zbMATH DE Number: 4165782
zbMATH Keywords: canonical Jordan form for matrices
zbMATH Keywords: bounded operators on infinite dimensional Hilbert spaces
zbMATH Keywords: spectral operators
zbMATH Keywords: decomposition for operators
zbMATH Keywords: sufficiently many invariant subspaces
zbMATH Keywords: finitely additive spectral measure
zbMATH Keywords: nest of invariant subspaces
zbMATH Keywords: Volterra operators
zbMATH Keywords: Riesz elements in \(C^ *\)-algebras modulo closed ideals
reviewed by: Mihai Putinar
MaRDI profile type: Publication
full work available at URL: 0022-1236(90)90122-2
OpenAlex ID: W1979874768
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description: scientific article; zbMATH DE number 4165782