Reverses of the Golden-Thompson type inequalities due to Ando-Hiai-Petz
| Autor: | Yuki Seo |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2008 |
| Předmět: |
15A48
Algebra and Number Theory unitarily invariant norm Positive-definite matrix generalized Kantorovich constant 15A42 15A45 Golden–Thompson inequality Hermitian matrix Combinatorics Algebra positive semidefinite matrix geometric mean Norm (mathematics) 15A60 reverse inequality Mond-Pecaric method Golden-Thompson inequality Specht ratio Analysis Mathematics |
| Zdroj: | Banach J. Math. Anal. 2, no. 2 (2008), 140-149 |
| Popis: | 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 $Y$. |
| Databáze: | OpenAIRE |
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