Some remarks on the spectral theory and commutativity of unbounded normal operators
type: Item
instance of: scholarly article
title: Some remarks on the spectral theory and commutativity of unbounded normal operators
zbMATH Open document ID: 1318.47105
author: Fotios C. Paliogiannis
published in: Complex Analysis and Operator Theory
publication date: 2014-12-17T00:00:00Z
review text: In this paper, some standard results from the spectral theory and commutativity of unbounded normal operators are proven, for instance, the spectral theorem, the characterization of the spectrum and the spectral mapping theorem. Nonetheless, it seems that some parts of the proofs are new. As an implication of the proven results, a theorem concerning commuting powers of normal operators is presented. More precisely, the following is proven. Theorem. If \(A\) is an unbounded normal operator and \(B\) is a bounded normal operator such that the spectrum \(\sigma(A)\) is generalized irrotational \((\mathrm{mod}(\frac{2\pi}{n}))\) -- for the definition see the paper -- and \(\sigma(B)\) is generalized irrotational \((\mathrm{mod}(\frac{2\pi}{m}))\), then \(B^m A^n \subseteq A^n B^m\) gives \(BA\subseteq AB\).
reviewed by: Janko Bračič
Mathematics Subject Classification ID: 47L60
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 47C15
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 6381415
zbMATH Keywords: Stonean space
zbMATH Keywords: abelian von Neumann algebra
zbMATH Keywords: Gelfand isomorphism
zbMATH Keywords: unbounded Borel functional calculus
zbMATH Keywords: spectral theorem
zbMATH Keywords: commuting powers
MaRDI profile type: Publication
full work available at URL: s11785-013-0325-3
OpenAlex ID: W2039360898
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DOI: 10.1007/S11785-013-0325-3
description: scientific article; zbMATH DE number 6381415