Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 241908, 12 pages
Research Article

Trace-Inequalities and Matrix-Convex Functions

Hokkaido University (Emeritus), Shiroishi-ku, Hongo-dori 9, Minami 4-10-805, Sapporo 003-0024, Japan

Received 8 October 2009; Accepted 30 November 2009

Academic Editor: Anthony To Ming Lau

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.


A real-valued continuous function f(t) on an interval (α,β) gives rise to a map Xf(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))(CB)Tr(f(C)f(B))(BA) for ABC. A related topic will be also discussed.