On reverses of the Golden-Thompson type inequalities

Abstract

In this paper we present some reverses of the Golden-Thompson type inequalities: Let $H$ and $K$ be Hermitian matrices such that $e^s{e}^H{⪯}_{ols}{e}^K{⪯}_{ols}{e}^t{e}^H$ for some scalars $s\le t$, and $\alpha \in [0,1]$. Then for all $p>0$ and $k=1,2,\dots ,n$ \begin{align*} \label{} λ_k (e^{(1-α)H + αK} ) \leq (\max \lbrace S(e^{sp}), S(e^{tp})\rbrace)^{\frac{1}{p}} λ_k (e^{pH} \sharp_αe^{pK})^{\frac{1}{p}}, \end{align*} where $A{♯}_{\alpha }B=A^{\frac{1}{2}}(A^{-\frac{1}{2}}{B}^{\frac{1}{2}}{A}^{-\frac{1}{2}}{)}^{\alpha }{A}^{\frac{1}{2}}$ is $\alpha$-geometric mean, $S(t)$ is the so called Specht's ratio and ${⪯}_{ols}$ is the so called Olson order. The same inequalities are also provided with other constants. The obtained inequalities improve some known results.