Operator entropy inequalities

M. S. Moslehian, F. Mirzapour, and A. Morassaei. "Operator entropy inequalities." Colloquium Mathematicae 130.2 (2013): 159-168. <http://eudml.org/doc/284381>.

@article{M2013,
abstract = {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.},
author = {M. S. Moslehian, F. Mirzapour, A. Morassaei},
journal = {Colloquium Mathematicae},
keywords = {-divergence functional; Jensen inequality; operator entropy; entropy inequality; operator concavity; perspective function; positive linear map},
language = {eng},
number = {2},
pages = {159-168},
title = {Operator entropy inequalities},
url = {http://eudml.org/doc/284381},
volume = {130},
year = {2013},
}

TY - JOUR
AU - M. S. Moslehian
AU - F. Mirzapour
AU - A. Morassaei
TI - Operator entropy inequalities
JO - Colloquium Mathematicae
PY - 2013
VL - 130
IS - 2
SP - 159
EP - 168
AB - 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.
LA - eng
KW - -divergence functional; Jensen inequality; operator entropy; entropy inequality; operator concavity; perspective function; positive linear map
UR - http://eudml.org/doc/284381
ER -