On a new construction of geometric mean of \(n\)-operators
type: Item
instance of: scholarly article
title: On a new construction of geometric mean of \(n\)-operators
zbMATH Open document ID: 1172.47017
author: Changdo Jung
author: Takeaki Yamazaki
author: Hosoo Lee
published in: Linear Algebra and its Applications
publication date: 2009-09-14T00:00:00Z
review text: \textit{W.\,Pusz} and \textit{S.\,L.\thinspace Woronowicz} [Rep.\ Math.\ Phys.\ 8, 159--170 (1975; Zbl 0327.46032)] introduced the geometric mean of two positive invertible operators \(A\) and \(B\) as \[ A \sharp B = A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}. \] One cannot immediately use this definition for defining the geometric mean of \(n\) operators. \textit{T.\,Ando}, \textit{C.\,K.\thinspace Li} and \textit{R.\,Mathias} [Linear Algebra Appl.\ 385, 305--334 (2004; Zbl 1063.47013)] defined the geometric mean of \(n\) positive definite operators by a symmetric procedure. Their definition requires an enormous amount of calculation. In the paper under review, the authors introduce a new construction of the geometric mean of \(n\) operators, which can be obtained easier than the geometric mean of Ando--Li--Mathias.
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 47A64
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 47L25
zbMATH DE Number: 5603566
zbMATH Keywords: operator inequality
zbMATH Keywords: positive operator
zbMATH Keywords: Thompson metric
zbMATH Keywords: geometric mean
MaRDI profile type: Publication
full work available at URL: j.laa.2009.05.012
OpenAlex ID: W1994169084
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DOI: 10.1016/J.LAA.2009.05.012
description: scientific article; zbMATH DE number 5603566