Colloquium Mathematicum, ( ISI ), Volume (130), No (2), Year (2013-10) , Pages (159-168)

#### Title : ( Operator entropy inequalities )

Authors: Mohammad Sal Moslehian , F. Mirzapour , A. Morassaei ,

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#### Abstract

In this paper we investigate a notion, related to $f$-divergence functional, 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 $\\\\mathbf{A}=(A_1,\\\\cdots,A_n)$ and $\\\\mathbf{B}=(B_1,\\\\cdots,B_n)$ 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 $$S_q^f(\\\\mathbf{A}|\\\\mathbf{B}):=\\\\sum_{j=1}^nA_j^{\\\\frac{1}{2}}\\\\left(A_j^{-\\\\frac{1}{2}}B_jA_j^{-\\\\frac{1}{2}}\\\\right)^qf\\\\left(A_j^{-\\\\frac{1}{2}}B_jA_j^{-\\\\frac{1}{2}}\\\\right)A_j^{\\\\frac{1}{2}}\\\\,,$$ 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. Afterwards, some inequalities concerning the classical Shannon entropy are drawn from it.

#### Keywords

, Information theory, f-divergence functional, Jensen inequality, entropy inequality, operator concavity, perspective function, positive linear map
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@article{paperid:1029102,
author = {Sal Moslehian, Mohammad and F. Mirzapour and A. Morassaei},
title = {Operator entropy inequalities},
journal = {Colloquium Mathematicum},
year = {2013},
volume = {130},
number = {2},
month = {October},
issn = {0010-1354},
pages = {159--168},
numpages = {9},
keywords = {Information theory; f-divergence functional; Jensen inequality; entropy inequality; operator concavity; perspective function; positive linear map},
}

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%0 Journal Article
%T Operator entropy inequalities
%A Sal Moslehian, Mohammad
%A F. Mirzapour
%A A. Morassaei
%J Colloquium Mathematicum
%@ 0010-1354
%D 2013

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