Matrix power means and new characterizations of operator monotone functions

Abstract

For positive definite matrices $A$ and $B$, the Kubo-Ando matrix power mean is defined as $$P_{\mu }(p,A,B)=A^{1/2}{\left(\frac{1+(A^{-1/2}B{A}^{-1/2}{)}^p}{2}\right)}^{1/p}{A}^{1/2}(p\ge 0).$$ In this paper, for $0\le p\le 1\le q$, we show that if one of the following inequalities \begin{align*} f(P_μ(p, A, B)) \le f(P_μ(1, A, B)) \le f(P_μ(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.

13 pages, some typos are fixed