Functional analysis
Relative entropy and Tsallis entropy of two accretive operators
Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 687-693.

Let A and B be two accretive operators. We first introduce the weighted geometric mean of A and B together with some related properties. Afterwards, we define the relative entropy as well as the Tsallis entropy of A and B. The present definitions and their related results extend those already introduced in the literature for positive invertible operators.

Soient A et B deux opérateurs accrétifs. Nous introduisons d'abord une moyenne géométrique pondérée de A et de B et nous en étudions certaines propriétés. Nous définissons ensuite l'entropie relative ainsi que l'entropie de Tsallis de A et de B. Ces définitions et les résultats obtenus étendent ceux déjà énoncés dans la littérature pour les opérateurs inversibles positifs.

Accepted:
Published online:
DOI: 10.1016/j.crma.2017.05.005

Mustapha Raïssouli 1, 2; Mohammad Sal Moslehian 3; Shigeru Furuichi 4

1 Department of Mathematics, Science Faculty, Taibah University, Al Madinah Al Munawwarah, P.O. Box 30097, Zip Code 41477, Saudi Arabia
2 Department of Mathematics, Faculty of Science, Moulay Ismail University, Meknes, Morocco
3 Department of Pure Mathematics, P.O. Box 1159, Ferdowsi University of Mashhad, Mashhad 91775, Iran
4 Department of Information Science, College of Humanities and Sciences, Nihon University, 3-25-40, Sakurajyousui, Setagaya-ku, Tokyo, 156-8550, Japan
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Mustapha Raïssouli; Mohammad Sal Moslehian; Shigeru Furuichi. Relative entropy and Tsallis entropy of two accretive operators. Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 687-693. doi : 10.1016/j.crma.2017.05.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.05.005/

[1] S. Drury Principal powers of matrices with positive definite real part, Linear Multilinear Algebra, Volume 63 (2015) no. 2, pp. 296-301

[2] J.I. Fujii; E. Kamei Relative operator entropy in noncommutative information theory, Math. Jpn., Volume 34 (1989) no. 3, pp. 341-348

[3] J.I. Fujii; Y. Seo Tsallis relative operator entropy with negative parameters, Adv. Oper. Theory, Volume 1 (2016) no. 2, pp. 219-235

[4] S. Furuichi Inequalities for Tsallis relative entropy and generalized skew information, Linear Multilinear Algebra, Volume 59 (2011) no. 10, pp. 1143-1158

[5] T. Furuta Parametric extensions of Shannon inequality and its reverse one in Hilbert space operators, Linear Algebra Appl., Volume 381 (2004), pp. 219-235

[6] O. Hirzallah; F. Kittaneh; M. Krnić; N. Lovričenvić; J. Pečarić Refinements and reverses of means inequalities for Hilbert space operators, Banach J. Math. Anal., Volume 7 (2013) no. 2, pp. 15-29

[7] F. Kittaneh; M. Krnić; N. Lovrčenvić; J. Pečarić Improved arithmetic–geometric and Heinz means inequalities for Hilbert space operators, Publ. Math. (Debr.), Volume 80 (2012) no. 3–4, pp. 465-478

[8] M. Lin Some inequalities for sector matrices, Oper. Matrices, Volume 10 (2016) no. 4, pp. 915-921

[9] M. Lin; F. Sun A property of the geometric mean of accretive operator, Linear Multilinear Algebra, Volume 65 (2017) no. 3, pp. 433-437

[10] R. Mathias Matrices with positive definite Hermitian part: inequalities and linear systems, SIAM J. Matrix Anal. Appl., Volume 13 (1992) no. 2, pp. 640-654

[11] M.S. Moslehian; F. Mirzapour; A. Morassaei Operator entropy inequalities, Colloq. Math., Volume 2548 (2013) no. 2, pp. 159-168

[12] R. Pal; M. Singh; M.S. Moslehian; J.S. Aujla A new class of operator monotone functions via operator means, Linear Multilinear Algebra, Volume 64 (2016) no. 12, pp. 2463-2473

[13] M. Raïssouli Some inequalities involving quadratic forms of operator means and operator entropies, Linear Multilinear Algebra (2017) (in press)

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