A generalization of Araki's log-majorization

摘要

We generalize Araki's log-majorization to the log-convexity theorem for the eigenvalues of Phi(A(p))(1/2) Psi(B-p) Phi(A(p))(1/2) as a function of p >= 0, where A, B are positive semidefinite matrices and Phi, Psi are positive linear maps between matrix algebras. Similar generalizations of the log-majorization of Ando-Hiai type for the weighted geometric mean A #(alpha) B are also given, including a lemma on general operator means A sigma B of A and a projection E. (C) 2016 Elsevier Inc. All rights reserved.