Reverses of the Golden-Thompson type inequalities due to Ando-Hiai-Petz

Abstract

In this paper, we show reverses of the Golden-Thompson type inequalities due to Ando, Hiai and Petz: Let $H$ and $K$ be Hermitian matrices such that $mI\leq H,K\leq MI$ for some scalars $m\leq M$, and let $\alpha \in [0,1]$. Then for every unitarily invarint norm \begin{equation*} |\! \| e^{(1-\alpha)H+\alpha K} |\! \| \ \leq \ S(e^{p(M-m)})^{\frac{1}{p}} \ |\! \| \left( e^{pH}\ \sharp _{\alpha} \ e^{pK} \right)^{\frac{1}{p}} |\! \| \end{equation*} holds for all positive number $p$ and the right-hand side converges to the left-hand side as $p\downarrow 0$, where $S(a)$ is the Specht ratio and the $\alpha$-geometric mean $X \ \sharp_{\alpha} \ Y$ is defined as \[ X\ \sharp _{\alpha} \ Y = X^{\frac{1}{2}} \left( X^{-\frac{1}{2}}YX^{-\frac{1}{2}} \right) ^{\alpha} X^{\frac{1}{2}} for all 0\leq \alpha \leq 1 \] for positive definite $X$ and $Y$.