Generalization of Ky Fan-Amir-Moez-Horn-Mirsky's result on the eigenvalues and real singular values of a matrix

Abstract

Ky Fan's result states that the real part of the eigenvalues of an n by n com- plex matrix A is majorized by the eigenvalues of the Hermitian part of A. The converse was established by Amir-Moez and Horn, and Mirsky, independently. We extend the results in the context of complex semisimple Lie algebras. Inequalities associated with the classical complex Lie algebras are given. The real case is also discussed.