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2015 Order of operators determined by operator mean
Masaru Nagisa, Mitsuru Uchiyama
Tohoku Math. J. (2) 67(1): 39-50 (2015). DOI: 10.2748/tmj/1429549578

Abstract

Let $\sigma$ be an operator mean and $f$ a non-constant operator monotone function on $(0,\infty)$ associated with $\sigma$. If operators $A, B$ satisfy $0\le A \le B$, then it holds that $Y \sigma (tA+X) \le Y \sigma (tB+X)$ for any non-negative real number $t$ and any positive, invertible operators $X,Y$. We show that the condition $ Y \sigma (tA+X) \le Y \sigma (tB+X)$ for a sufficiently small $t>0$ implies $A \le B$ if and only if $X$ is a positive scalar multiple of $Y$ or the associated operator monotone function $f$ with $\sigma$ has the form $f(t) = (at+b)/(ct+d)$, where $a,b,c,d$ are real numbers satisfying $ad-bc>0$.

Citation

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Masaru Nagisa. Mitsuru Uchiyama. "Order of operators determined by operator mean." Tohoku Math. J. (2) 67 (1) 39 - 50, 2015. https://doi.org/10.2748/tmj/1429549578

Information

Published: 2015
First available in Project Euclid: 20 April 2015

zbMATH: 1326.47017
MathSciNet: MR3337962
Digital Object Identifier: 10.2748/tmj/1429549578

Subjects:
Primary: 47A63
Secondary: 15A39

Rights: Copyright © 2015 Tohoku University

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