New characterizations of operator monotone functions

If$\sigma$is a symmetric mean and$f$is an operator monotone function on$[0, \infty)$ , then$$f(2(A^{-1}+B^{-1})^{-1})\le f(A\sigma B)\le f((A+B)/2).$$Conversely, Ando and Hiai showed that if$f$is a function that satisfies either one of these inequalities for all positive operators$A$and$B$and a symm...

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Bibliographic Details
Main Authors Dinh, Trung Hoa, Dumitru, Raluca, Franco, Jose
Format Journal Article
LanguageEnglish
Published 18.03.2018
Subjects
Mathematics - Functional Analysis
Mathematics - Operator Algebras
Online AccessGet full text
DOI10.48550/arxiv.1803.06659

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Summary:If$\sigma$is a symmetric mean and$f$is an operator monotone function on$[0, \infty)$ , then$$f(2(A^{-1}+B^{-1})^{-1})\le f(A\sigma B)\le f((A+B)/2).$$Conversely, Ando and Hiai showed that if$f$is a function that satisfies either one of these inequalities for all positive operators$A$and$B$and a symmetric mean different than the arithmetic and the harmonic mean, then the function is operator monotone. In this paper, we show that the arithmetic and the harmonic means can be replaced by the geometric mean to obtain similar characterizations. Moreover, we give characterizations of operator monotone functions using self-adjoint means and general means subject to a constraint due to Kubo and Ando.
DOI:10.48550/arxiv.1803.06659