Operator log-convex functions and f-divergence functional

TL;DR

This paper characterizes operator log-convex functions through positive linear mappings and explores their non-commutative f-divergence functional, establishing conditions for operator log-convexity and convexity.

Contribution

It provides a new characterization of operator log-convex functions using positive linear mappings and analyzes their non-commutative f-divergence functional.

Findings

f is operator log-convex iff the non-commutative f-divergence functional is operator log-convex in the first variable

f is operator log-convex iff the non-commutative f-divergence functional is operator convex in the second variable

The paper establishes a precise equivalence condition for operator log-convexity

Abstract

We present a characterization of operator log-convex functions by using positive linear mappings. Moreover, we study the non-commutative f-divergence functional of operator log-convex functions. In particular, we prove that f is operator log-convex if and only if the non-commutative f-divergence functional is operator log-convex in its first variable and operator convex in its second variable.