Relations between two operator inequalities motivated by the theory of operator means
type: Item
instance of: scholarly article
title: Relations between two operator inequalities motivated by the theory of operator means
zbMATH Open document ID: 1120.47011
author: Masatoshi Ito
published in: Integral Equations and Operator Theory
publication date: 2006-02-08T00:00:00Z
review text: The author shows several results on operator inequalities motivated by the theory of operator means. The main result is the following Theorem. Let \(A\) and \(B\) be positive operators, and let \(f\) and \(g\) be non-negative continuous functions on \([0,\infty)\) satisfying \(f(t)g(t)=t\). Then the following hold: (i) \(f(B^\frac{1}{2}AB^\frac{1}{2})\geq B\) ensures that \(A-g(A^\frac{1}{2}BA^\frac{1}{2})\geq A^\frac{1}{2}E_BA^\frac{1}{2}-g(0)E_{A^\frac{1}{2} BA^\frac{1}{2}},\) (ii) \(B\geq f(B^\frac{1}{2}AB^\frac{1}{2})\) ensures that \(g(A^\frac{1}{2}BA^\frac{1}{2})-A\geq g(0))E_{A^\frac{1}{2} BA^\frac{1}{2}}-A^\frac{1}{2}E_BA^\frac{1}{2},\) where \(E_B\) and \(E_{A^\frac{1}{2} BA^\frac{1}{2}}\) are the orthoprojections to the kernels of \(B\) and \(A^\frac{1}{2}BA^\frac{1}{2}\), respectively.
reviewed by: Aleksej Turnšek
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 47A64
zbMATH DE Number: 5005105
zbMATH Keywords: operator inequality
zbMATH Keywords: operator mean
zbMATH Keywords: representing function
zbMATH Keywords: positive operators
MaRDI profile type: Publication
full work available at URL: s00020-004-1321-9
OpenAlex ID: W2044413631
DOI: 10.1007/S00020-004-1321-9
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description: scientific article; zbMATH DE number 5005105