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2008 Reverses of the Golden-Thompson type inequalities due to Ando-Hiai-Petz
Yuki Seo
Banach J. Math. Anal. 2(2): 140-149 (2008). DOI: 10.15352/bjma/1240336300


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 $Y$.


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Yuki Seo. "Reverses of the Golden-Thompson type inequalities due to Ando-Hiai-Petz." Banach J. Math. Anal. 2 (2) 140 - 149, 2008.


Published: 2008
First available in Project Euclid: 21 April 2009

zbMATH: 1188.15018
MathSciNet: MR2436874
Digital Object Identifier: 10.15352/bjma/1240336300

Primary: 15A42
Secondary: 15A45 , 15A48 , 15A60

Keywords: generalized Kantorovich constant , geometric mean , Golden-Thompson inequality , Mond-Pecaric method , positive semidefinite matrix , reverse inequality , Specht ratio , ‎unitarily invariant norm

Rights: Copyright © 2008 Tusi Mathematical Research Group

Vol.2 • No. 2 • 2008
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