Shannon like inequalities for f-connections of positive linear maps and positive operators

Abstract

In this paper, we investigate a map (T, B) → T fT − (B), called an f -connection, induced by an operator convex (concave) function f , where T − denotes a reflexive generalized inverse of a positive linear map T between unital C * -algebras A and B of Hilbert space operators, and B ∈ B. Some special cases of f -connection with invertible T are related to geometric operator mean, relative operator entropy and Csiszár operator f -divergence. We formulate conditions under which the inequality f (1)I ≤ n k=1 T k fT − k (B k ) holds, where T k : A → B are positive linear maps and B k ∈ Ran T k . In particular, the following Shannon like inequality 0 ≤ n k=1 T k fT − k (B k ) is shown for an operator convex function f with f (1) = 0. The obtained results are specified for Csiszár operator f -divergence. Some recent results by Isa et al. (2013) [15] are recovered.