Operator entropy inequalities

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 and then give upper and lower bounds for Sfq (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. ©Instytut Matematyczny PAN, 2013.