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Reverses of the Golden-Thompson type inequalities due to Ando-Hiai-Petz. (English) Zbl 1188.15018

Summary: We show reverses of the Golden-Thompson type inequalities due to T. Ando and F. Hiai [Linear Algebra Appl. 197–198, 113–131 (1994; Zbl 0793.15011)] and to F. Hiai and D. Petz [Linear Algebra Appl. 181, 153–185 (1993; Zbl 0784.15011)]: 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 invariant norm \[ \||e^{(1-\alpha)H+\alpha K}\||\leq S(e^{p(M-m)})^{\frac 1p}\||(e^{pH}\#_\alpha e^{pK})^{\frac 1p}\|| \]
holds for all \(p > 0\) 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 \(X\,\#_\alpha\, Y\) is defined as
\[ X\,\#_\alpha\, Y = X^{\frac12}(X^{-\frac12} YX^{-\frac12})^\alpha X^{\frac12}\text{ for all }0\leq\alpha\leq 1 \]
for positive definite \(X\) and \(Y\).

MSC:

15A45 Miscellaneous inequalities involving matrices
15A42 Inequalities involving eigenvalues and eigenvectors
15B48 Positive matrices and their generalizations; cones of matrices
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15B57 Hermitian, skew-Hermitian, and related matrices
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