Powers and commutativity of selfadjoint operators
type: Item
instance of: scholarly article
title: Powers and commutativity of selfadjoint operators
zbMATH Open document ID: 0839.47015
DOI: 10.1007/BF01450506
author: Mitsuru Uchiyama
published in: Mathematische Annalen
publication date: 1996-06-24T00:00:00Z
full work available at URL: https://eudml.org/doc/165272
review text: Let \(B({\mathfrak H})\) be the algebra of bounded operators on a Hilbert space \(\mathfrak H\). An operator \(T\in B({\mathfrak H})\) is said to be accretive if \(\text{Re } T\geq 0\). It has been shown by \textit{C. R. DePrima} and \textit{B. K. Richard} [Indiana Univ. Math. J. 23, 163-172 (1973/74; Zbl 0257.47028)] that if \(T^n\) is accretive for \(n= 1,2,\dots\), then \(T\geq 0\). In the present paper this theorem is extended to pairs of operators, and some applications are given to proving commutativity of selfadjoint operators. For example, it is shown that if \(X,Y\in B({\mathfrak H})\), if \(\text{Re } X\geq \text{Re } Y\geq 0\), and if \(X^n+ Y^n\geq 0\) for \(n= 1, 2,\dots\), then \(X\geq Y\geq 0\). As an application of this result it is shown that if \(A\), \(B\) and \(C\) are selfadjoint operators for which \(B\geq C\), if \(AB^n+ C^n A\geq 0\) for \(n= 1, 2,\dots\), and if \(A\geq 0\) or \(C\geq 0\), then \(A\) commutes with \(B\) and \(C\).
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 47B44
Mathematics Subject Classification ID: 47B65
zbMATH DE Number: 734591
zbMATH Keywords: positive operator
zbMATH Keywords: algebra of bounded operators
zbMATH Keywords: accretive
zbMATH Keywords: commutativity
zbMATH Keywords: selfadjoint operators
reviewed by: Richard C. Gilbert
MaRDI profile type: Publication
cites work: Majorization, doubly stochastic matrices, and comparison of eigenvalues
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description: scientific article; zbMATH DE number 734591