Operator entropy inequalities

M. S. Moslehian ¹   ; F. Mirzapour ²   ; A. Morassaei ²

¹ Department of Pure Mathematics Center of Excellence in Analysis on Algebraic Structures Ferdowsi University of Mashhad P.O. Box 1159 Mashhad 91775, Iran
² Department of Mathematics Faculty of Sciences University of Zanjan P.O. Box 45195-313 Zanjan, Iran

Colloquium Mathematicum, Tome 130 (2013) no. 2, pp. 159-168

Résumé

We investigate a notion of relative operator entropy, which develops the theory started by J. I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341–348]. For two finite sequences $\mathbf{A}=(A_1,\ldots,A_n)$ and $\mathbf{B}=(B_1,\ldots,B_n)$ of positive operators acting on a Hilbert space, a real number $q$ and an operator monotone function $f$ we extend the concept of entropy by setting $$\def\mfrac#1#2{#1/#2} S_q^f(\mathbf{A}\,|\,\mathbf{B}):=\sum_{j=1}^nA_j^{\mfrac{1}{2}} (A_j^{-\mfrac{1}{2}}B_jA_j^{-\mfrac{1}{2}})^qf(A_j^{-\mfrac{1}{2}}B_jA_j^{-\mfrac{1}{2}})A_j^{\mfrac{1}{2}} , $$ and then give upper and lower bounds for $S_q^f(\mathbf{A}\,|\,\mathbf{B})$ as an extension of an inequality due to T. Furuta [Linear Algebra Appl. 381 (2004), 219–235] under certain conditions. As an application, some inequalities concerning the classical Shannon entropy are deduced.

Détail

DOI : 10.4064/cm130-2-2

Keywords: investigate notion relative operator entropy which develops theory started fujii kamei math japonica finite sequences mathbf ldots mathbf ldots positive operators acting hilbert space real number operator monotone function extend concept entropy setting def mfrac mathbf mathbf sum mfrac mfrac mfrac mfrac mfrac mfrac upper lower bounds mathbf mathbf extension inequality due furuta linear algebra appl under certain conditions application inequalities concerning classical shannon entropy deduced