INEQUALITIES FOR THE DUAL RELATIVE OPERATOR ENTROPY

Abstract

In this paper, we introduce the concept of dual relative entropy defined by \(D(A\mid B):=A^{1/2}\left(A^{-1/2}BA^{-1/2}\ln\left(A^{-1/2}BA^{-1/2}\right)\right)A^{1/2}\) for positive invertible operators \(A\) and \(B\) and establish various upper and lower bounds for the error operator in approximating the \(D(A\mid B)\) by \(\frac{m\ln m}{M-m}(MA-B)+\frac{M\ln M}{M-m}(B-mA)\) under the natural assumption \(mA\leq B\leq MA\) for some \(m,M\) with \(0<m<M\). Applications for the operator entropy are also given. Some trace inequalities are derived as well.