Lower and Upper Bounds for the Generalized Csiszár f-divergence Operator Mapping

Let \({\textbf{A}}=\{A_{1},...,A_{n}\}\) and \({\textbf{B}}=\{B_{1},...,B_{n}\}\) be two finite sequences of strictly positive operators on a Hilbert space \( {\mathcal {H}}\) and f, \(h:{\mathbb {I}}\rightarrow {\mathbb {R}}\) continuous functions with \(h>0\).. We consider the generalized Csiszár f-divergence operator mapping defined by

$$\begin{aligned} {\textbf{I}}_{f\Delta h}({\textbf{A}},{\textbf{B}})=\sum _{i=1}^{n}P_{f\Delta h}(A_{i},B_{i}), \end{aligned}$$

where

$$\begin{aligned} P_{f\Delta h}(A,B):=h(A)^{1/2}f(h(A)^{-1/2}Bh(A)^{-1/2})h(A)^{1/2} \end{aligned}$$

is introduced for every strictly positive operator A and every self-adjoint operator B, where the spectrum of the operators

$$\begin{aligned} A, A^{-1/2}BA^{-1/2}\text { and }h(A)^{-1/2}Bh(A)^{-1/2} \end{aligned}$$

are contained in the closed interval \({\mathbb {I}}\). In this paper we obtain some lower and upper bounds for \({\textbf{I}}_{f\Delta h}({\textbf{A}},{\textbf{B}})\) with applications to the geometric operator mean and the relative operator entropy. We verify the information monotonicity for the Csisz ár f-divergence operator mapping and the generalized Csiszár f-divergence operator mapping.