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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 A = (A₁,...,Aₙ) and B = (B₁,...,Bₙ) 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
$S_q^f(A|B): = ∑_{j=1}^{n} A_j^{1/2} (A_j^{-1/2}B_jA_j^{-1/2})^{q} f(A_j^{-1/2}B_jA_j^{-1/2})A_j^{1/2}$,
and then give upper and lower bounds for $S_q^f(A|B)$ 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.
$S_q^f(A|B): = ∑_{j=1}^{n} A_j^{1/2} (A_j^{-1/2}B_jA_j^{-1/2})^{q} f(A_j^{-1/2}B_jA_j^{-1/2})A_j^{1/2}$,
and then give upper and lower bounds for $S_q^f(A|B)$ 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.
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Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
159-168
Opis fizyczny
Daty
wydano
2013
Twórcy
autor
- Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran
autor
- Department of Mathematics, Faculty of Sciences, University of Zanjan, P.O. Box 45195-313, Zanjan, Iran
autor
- Department of Mathematics, Faculty of Sciences, University of Zanjan, P.O. Box 45195-313, Zanjan, Iran
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bwmeta1.element.bwnjournal-article-doi-10_4064-cm130-2-2