Banach Journal of Mathematical Analysis

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

Yuki Seo

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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 $Y$.

Article information

Banach J. Math. Anal., Volume 2, Number 2 (2008), 140-149.

First available in Project Euclid: 21 April 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A42: Inequalities involving eigenvalues and eigenvectors
Secondary: 15A45: Miscellaneous inequalities involving matrices 15A48 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05]

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


Seo, Yuki. Reverses of the Golden-Thompson type inequalities due to Ando-Hiai-Petz. Banach J. Math. Anal. 2 (2008), no. 2, 140--149. doi:10.15352/bjma/1240336300.

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