Around Operator Monotone Functions

TL;DR

This paper characterizes when the symmetrized product of positive operators is positive using operator monotone functions, and explores conditions for the composition of operator convex and monotone functions to be operator monotone, with applications.

Contribution

It provides a new characterization of positivity for symmetrized products and establishes conditions for the composition of operator convex and monotone functions to be operator monotone.

Findings

Symmetrized product positivity characterized by operator monotone functions.

Necessary and sufficient conditions for composition of operator convex and monotone functions.

Derived operator inequalities and applications.

Abstract

We show that the symmetrized product $AB+BA$ of two positive operators $A$ and $B$ is positive if and only if $f(A+B)\leq f(A)+f(B)$ for all non-negative operator monotone functions $f$ on $[0,\infty)$ and deduce an operator inequality. We also give a necessary and sufficient condition for that the composition $f\circ g$ of an operator convex function $f$ on $[0,\infty)$ and a non-negative operator monotone function $g$ on an interval $(a,b)$ is operator monotone and give some applications.