Limits of strongly irreducible operators, and the Riesz decomposition theorem
type: Item
instance of: scholarly article
title: Limits of strongly irreducible operators, and the Riesz decomposition theorem
zbMATH Open document ID: 0734.47011
author: Domingo A. Herrero
author: Chunlan Jiang
published in: Michigan Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: Let \({\mathcal L}({\mathcal H})\) be the algebra of all bounded linear operators acting on a complex, separable, infinite dimensional Hilbert space \({\mathcal H}\). The class of strongly irreducible operators is \[ {\mathcal S}{\mathcal J}({\mathcal H})=\{T\in {\mathcal L}({\mathcal H}):\;T\quad does\quad not\quad commute\text{ with } any\quad nontrivial\quad idempotent\}. \] The authors prove that the norm closure of \({\mathcal S}{\mathcal J}({\mathcal H})\) is \(\{\) \(T\in {\mathcal L}({\mathcal H}):\sigma\) (T) is connected\(\}\). Here \(\sigma\) (T) is the spectrum of T. This theorem relates to the Riesz decomposition theorem which says that if \(\sigma\) (T) is not connected, T does commute with a nontrivial idempotent operator. The proof requires some careful constructions of operators which satisfy certain spectral properties.
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 47A53
zbMATH DE Number: 4214757
zbMATH Keywords: strongly irreducible operators
zbMATH Keywords: Riesz decomposition theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004135
OpenAlex ID: W2014206674
DOI: 10.1307/MMJ/1029004135
description: scientific article; zbMATH DE number 4214757