Two trace inequalities for operator functions

TL;DR

This paper establishes new trace inequalities for operator functions, demonstrating how the trace of differences relates to the trace of absolute differences under certain conditions, with implications for operator theory.

Contribution

It introduces two trace inequalities for operator monotone and convex functions, expanding understanding of operator function behavior in matrix analysis.

Findings

Proves an inequality for non-negative operator monotone functions with $f(0)=0$.

Shows the inequality reverses when $f$ is operator convex.

Provides a framework for comparing traces involving operator functions.

Abstract

In this paper we show that for a non-negative operator monotone function $f$ on $[0, \infty)$ such that $f(0)= 0$ and for any positive semidefinite matrices $A$ and $B$, $$ Tr((A-B)(f(A)-f(B))) \le Tr(|A-B|f(|A-B|)). $$ When the function $f$ is operator convex on $[0, \infty)$, the inequality is reversed.