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## Operator entropy inequalities

### Tom 130 / 2013

Colloquium Mathematicum 130 (2013), 159-168 MSC: Primary 47A63; Secondary 15A42, 46L05, 47A30. DOI: 10.4064/cm130-2-2

#### Streszczenie

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.

#### Autorzy

• M. S. MoslehianDepartment of Pure Mathematics
Center of Excellence in Analysis on Algebraic Structures
P.O. Box 1159
e-mail
e-mail
• F. MirzapourDepartment of Mathematics
Faculty of Sciences
University of Zanjan
P.O. Box 45195-313
Zanjan, Iran
e-mail
• A. MorassaeiDepartment of Mathematics
Faculty of Sciences
University of Zanjan
P.O. Box 45195-313
Zanjan, Iran
e-mail

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