Majorization and some operator monotone functions
type: Item
instance of: scholarly article
title: Majorization and some operator monotone functions
zbMATH Open document ID: 1188.47019
author: Mitsuru Uchiyama
published in: Linear Algebra and its Applications
publication date: 2010-03-04T00:00:00Z
review text: Let \(h\) be a non-decreasing function on an interval \(I\) and \(k\) be an increasing function on an interval \(J\). Then \(h\) is said to be majorized by \(k\), denoted by \(h \preceq k\,\,(J)\), if \(J\subseteq I\) and the composite \(h\circ k^{-1}\) is operator monotone on \(k(J)\). The main result of the paper is that \(\frac{t-a}{f(t)-f(a)}\) is a nonnegative operator monotone function if \(f(t)\) is operator monotone on \((0, \infty)\) and \(0 <a<\infty\). The author then utilizes the notion of majorization to conclude that \(\frac{(t-a)(t-b)}{(t^r-a^r )(t^{1-r}-b^{1-r})}\) is operator monotone on \([0,\infty)\) for \(0 \leq a, b < \infty\) and for \(0 \leq r \leq 1\). The special case where \(a = b = 1\) gives rise to a theorem of \textit{D.\,Petz} and \textit{H.\,Hasegawa} [Lett.\ Math.\ Phys.\ 38, No.\,2, 221--225 (1996; Zbl 0855.58070)].
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 15A39
zbMATH DE Number: 5677351
zbMATH Keywords: matrix order
zbMATH Keywords: operator monotone function
zbMATH Keywords: Löwner theorem
zbMATH Keywords: majorisation
MaRDI profile type: Publication
full work available at URL: j.laa.2008.11.023
OpenAlex ID: W1997527839
cites work: Q5690490
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DOI: 10.1016/J.LAA.2008.11.023
description: scientific article; zbMATH DE number 5677351