Operator entropy inequalities

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

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

Colloquium Mathematicae, Tome 131 (2013), p. 159-168 / Harvested from The Polish Digital Mathematics Library

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 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 $S_{q}^{f} (A | B) : = \sum_{j = 1}^{n} A_{j}^{1 / 2} {(A_{j}^{- 1 / 2} B_{j} A_{j}^{- 1 / 2})}^{q} f (A_{j}^{- 1 / 2} B_{j} A_{j}^{- 1 / 2}) A_{j}^{1 / 2}$ , 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.

Publié le : 2013-01-01

Zbl 1314.47025

EUDML-ID : urn:eudml:doc:284381

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     title = {Operator entropy inequalities},
     journal = {Colloquium Mathematicae},
     volume = {131},
     year = {2013},
     pages = {159-168},
     zbl = {1314.47025},
     language = {en},
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M. S. Moslehian; F. Mirzapour; A. Morassaei. Operator entropy inequalities. Colloquium Mathematicae, Tome 131 (2013) pp. 159-168. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm130-2-2/