A converse of Loewner-Heinz inequality and applications to operator means
type: Item
instance of: scholarly article
title: A converse of Loewner-Heinz inequality and applications to operator means
zbMATH Open document ID: 1322.47027
author: Mitsuru Uchiyama
author: Takeaki Yamazaki
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-03-27T00:00:00Z
review text: Let \(f\) be an operator monotone function. Then \(A \leq B\) implies \(f(A) \leq f(B)\), but the converse is not true. In this paper, the authors investigate the converse implication from the viewpoint of operator means. More precisely, they show that, if \(f(t )> 0\) is a non-constant operator monotone function defined on a neighborhood of \(t = 1\) with \(f(1) = 1\), and \(A\) and \(B\) are bounded self-adjoint operators, then the following statements are equivalent: {\parindent=0.8cm\begin{itemize}\item[(i)] \(A \leq B\), \item[(ii)] \(\lambda A + I \leq \lambda B + I\) for every \(\lambda \geq 0\), \item[(iii)] \(f(\lambda A + I)\leq f(\lambda B + I)\) for all sufficiently small \(\lambda \geq 0\), (iv) \(f(\lambda B + I)^{-1}\sharp f(\lambda A + I)\leq I\) for all sufficiently small \(\lambda \geq 0\), where \(\sharp\) denotes the usual operator mean. \end{itemize}} In addition, they prove that, if \(f\) is a non-constant operator monotone function on \((0,\infty)\) with \(f (1) = 1\), and \(A\) and \(B\) be bounded self-adjoint operators acting on a Hilbert space \(H\), then the following are equivalent: {\parindent=0.8cm\begin{itemize}\item[(i)] \(A \leq B\), \item[(ii)] \(\|x\|^2 \leq \|f(\lambda A+I)^{-1/2}x\|\,\|f(-\lambda B+I)^{-1/2}x\|\) for all \(x\in H\) and all sufficiently small \(\lambda \geq 0\), \item[(iii)] \(\|x\|^2 \leq \|e^{-pA}x\|\,\|e^{pB}x\|\) for all \(x\in H\) and all \(p \geq 0\). \end{itemize}} They also extend their results to the case of multi-variable power means and the Karcher mean.
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 47B65
zbMATH DE Number: 6420194
zbMATH Keywords: positive definite operators
zbMATH Keywords: Loewner-Heinz inequality
zbMATH Keywords: operator mean
zbMATH Keywords: operator monotone function
zbMATH Keywords: operator concave function
zbMATH Keywords: Ando-Hiai inequality
zbMATH Keywords: geometric mean
zbMATH Keywords: Karcher mean
zbMATH Keywords: power mean
MaRDI profile type: Publication
full work available at URL: j.jmaa.2013.11.055
OpenAlex ID: W2071568849
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DOI: 10.1016/J.JMAA.2013.11.055
description: scientific article; zbMATH DE number 6420194