Non‐commutative ‐divergence functional

Abstract

We introduce the non‐commutative f‐divergence functional urn:x-wiley:dummy:mana201200194:equation:mana201200194-math-0002for an operator convex function f, where and are continuous fields of Hilbert space operators and study its properties. We establish some relations between the perspective of an operator convex function f and the non‐commutative f‐divergence functional. In particular, an operator extension of Csiszár's result regarding f‐divergence functional is presented. As some applications, we establish a refinement of the Choi–Davis–Jensen operator inequality, obtain some unitarily invariant norm inequalities and give some results related to the Kullback–Leibler distance.