Characterizations of Operator Monotonicity via Operator Means and Applications to Operator Inequalities

TL;DR

This paper characterizes operator monotonicity using operator means, specifically the weighted harmonic mean, and derives new operator inequalities based on these characterizations.

Contribution

It provides novel characterizations of operator monotonicity via operator means and applies these to establish new operator inequalities.

Findings

Operator monotone increasing functions are characterized by inequalities involving the weighted harmonic mean.

Operator monotone decreasing functions are characterized by reverse inequalities.

New operator inequalities involving means are derived from these characterizations.

Abstract

We prove that a continuous function $f:(0,\infty) \to (0,\infty)$ is operator monotone increasing if and only if $f(A \: !_t \: B) \leqs f(A) \: !_t \: f(B)$ for any positive operators $A,B$ and scalar $t \in [0,1]$. Here, $!_t$ denotes the $t$-weighted harmonic mean. As a counterpart, $f$ is operator monotone decreasing if and only if the reverse of preceding inequality holds. Moreover, we obtain many characterizations of operator-monotone increasingness/decreasingness in terms of operator means. These characterizations lead to many operator inequalities involving means.