Power means of probability measures and Ando-Hiai inequality

Abstract

Let $\mu$ be a probability measure of compact support on the set ${\mathbb{P}}_n$ of all positive definite matrices, let $t\in (0,1]$, and let $P_t(\mu )$ be the unique positive solution of $X={\int }_{{\mathbb{P}}_n}X{♯}_{t}Zd\mu (Z)$. In this paper, we show that $$P_t(\mu )\le I⟹P_{\frac{t}{p}}(\nu )\le P_t(\mu )$$ for every $p\ge 1$, where $\nu (Z)=\mu (Z^{1/p})$. This provides an extension of the Ando--Hiai inequality for matrix power means. Moreover, we prove that if $\Phi :{\mathbb{M}}_n\to {\mathbb{M}}_m$ is a unital positive linear map, then $\Phi (P_t(\mu ))\le P_t(\nu )$ for all $t\in [-1,1]\backslash \{0\}$, where $\nu$ is a certain measure.

arXiv commentSubmitted on 24 November 2018