Trace-Inequalities and Matrix-Convex Functions

A real-valued continuous function [inline-graphic not available: see fulltext] on an interval [inline-graphic not available: see fulltext] gives rise to a map [inline-graphic not available: see fulltext] via functional calculus from the convex set of [inline-graphic not available: see fulltext] Hermitian matrices all of whose eigenvalues belong to the interval. Since the subpace of Hermitian matrices is provided with the order structure induced by the cone of positive semidefinite matrices, one can consider convexity of this map. We will characterize its convexity by the following trace-inequalities: [inline-graphic not available: see fulltext] for [inline-graphic not available: see fulltext]. A related topic will be also discussed.