Copyright © 2010 Tsuyoshi Ando. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A real-valued continuous function f(t) on an interval (α,β) gives rise to a map X↦f(X) via functional calculus from the convex set of n×n 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: Tr(f(B)−f(A))(C−B)≤Tr(f(C)−f(B))(B−A) for A≤B≤C. A related topic will be also discussed.