Matrix power means and new characterizations of operator monotone functions

TL;DR

This paper introduces new characterizations of operator monotone functions using matrix power means, extending the understanding of inequalities involving positive definite matrices and exploring inverse problems for non-Kubo-Ando means.

Contribution

It provides novel characterizations of operator monotone functions through inequalities involving matrix power means, including non-Kubo-Ando cases, and studies inverse problems for specific powers.

Findings

Operator monotonicity characterized by matrix power mean inequalities.

New inequalities involving non-Kubo-Ando matrix power means.

Characterizations of operator monotone functions via inverse problems.

Abstract

For positive definite matrices $A$ and $B$, the Kubo-Ando matrix power mean is defined as $$ P_\mu(p, A, B) = A^{1/2}\left(\frac{1+(A^{-1/2}BA^{-1/2})^p}{2}\right )^{1/p} A^{1/2}\quad (p \ge 0). $$ In this paper, for $0\le p \le 1 \le q$, we show that if one of the following inequalities \begin{align*} f(P_\mu(p, A, B)) \le f(P_\mu(1, A, B)) \le f(P_\mu(q, A, B))\nonumber \end{align*} holds for any positive definite matrices $A$ and $B$, then the function $f$ is operator monotone on $(0, \infty).$ We also study the inverse problem for non-Kubo-Ando matrix power means with the powers $1/2$ and $2$. As a consequence, we establish new charaterizations of operator monotone functions with the non-Kubo-Ando matrix power means.