Abstract
Ando et al. have proved that inequality \(\Re \mathfrak{e}trA^{p_1 } B^{q_1 \ldots } A^{p_k } B^{q_k } \leqslant trA^{p_1 + \ldots + p_k } B^{q_1 + \ldots + q_k }p\) is valid for all positive semidefinite matrices A,B and those nonnegative real numbers p 1, q 1,..., p k , q k which satisfy certain additional conditions. We give an example to show that this inequality is not valid for all collections of p 1, q 1,..., p k , q k ≥ 0. We also study related trace inequalities.
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References
T. Ando and F. Hiai, Log majorization and complementary Golden-Thompson type inequalities, Linear Algebra Appl., 198 (1994), 113–131.
T. Ando, F. Hiai and K. Okubo, Trace inequalities for multiple products of two matrices, Math. Inequal. Appl., 3 (2000), 307–318.
K. Audenaert, A singular value inequality for Heinz means, Linear Algebra Appl., 422 (2007), 279–283.
K. Audenaert, A norm inequality for pairs of commuting positive semidefinite matrices, Electron. J. Linear Algebra, 30 (2015), DOI: http://dx.doi.org/10.13001/1081-3810.2829.
R. Bhatia, Matrix Analysis, Springer, Berlin, 1997.
R. Bhatia, Trace inequalities for products of positive definite matrices, J. Math. Phys., 55 (2014).
T. Bottazzi, R. Elencwajg, G. Larotonda and A. Varela, Inequalities related to Bourin and Heinz means with a complex parameter, J. Math. Anal. Appl., 426 (2015), 765–773.
J.-C. Bourin, Some inequalities for norms on matrices and operators, Linear Algebra Appl., 292 (1999), 139–154.
J.-C. Bourin, Matrix versions of some classical inequalities, Linear Algebra Appl., 416 (2006), 890–907.
J.-C. Bourin, Matrix subadditivity inequalities and block-matrices, Int. J. Math., 20 (2009), 679–691.
S. Furuichi, K. Kuriyama and K. Yanagi, Trace inequalities for products of matrices, Linear Algebra Appl., 430 (2009), 2271–2276.
S. Hayajneh and F. Kittaneh, Lieb-Thirring trace inequalities and a question of Bourin, J. Math. Phys., 54, 033504 (2013); doi: 10.1063/1.4793993.
S. Hayajneh and F. Kittaneh, Trace inequalities and a question of Bourin, Bull. Aust. Math. Soc., 88 (2013), 384–389.
F. Hiai and X. Zhan, Inequalities involving unitarily invariant norms and operator monotone functions, Linear Algebra Appl., 341 (2002), 151–169.
C. J. Hillar and C. R. Johnson, Eigenvalues of words in two positive definite letters, SIAM J. Matrix Anal. Appl., 23 (2002), 916–928.
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Plevnik, L. On a matrix trace inequality due to Ando, Hiai and Okubo. Indian J Pure Appl Math 47, 491–500 (2016). https://doi.org/10.1007/s13226-016-0180-9
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DOI: https://doi.org/10.1007/s13226-016-0180-9