New characterizations of operator monotone functions

TL;DR

This paper provides new characterizations of operator monotone functions using geometric and other means, extending previous inequalities and offering deeper insights into their properties.

Contribution

It introduces novel characterizations of operator monotone functions by replacing classical means with geometric and self-adjoint means, broadening the theoretical framework.

Findings

Characterizations using geometric mean analogous to arithmetic and harmonic means.

Extension of inequalities to include self-adjoint and general means.

New criteria for operator monotonicity based on mean inequalities.

Abstract

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.