Operator entropy inequalities
Volume 130 / 2013
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 $\mathbf{A}=(A_1,\ldots,A_n)$ 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.
