A generalization of Araki's log-majorization

TL;DR

This paper extends Araki's log-majorization to a broader log-convexity theorem involving eigenvalues of matrix functions, with applications to positive linear maps and matrix inequalities.

Contribution

It introduces a generalized log-convexity theorem for eigenvalues of matrix functions, expanding Araki's and Ando-Hiai's log-majorization results.

Findings

Established a log-convexity theorem for eigenvalues of matrix functions involving positive linear maps.

Generalized Araki's log-majorization to a broader class of matrix inequalities.

Extended the log-majorization framework to include new inequalities for positive semidefinite matrices.

Abstract

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 $p\ge0$, where $A,B$ are positive semidefinite matrices and $\Phi,\Psi$ are positive linear maps between matrix algebras. A similar generalization of the log-majorization of Ando-Hiai type is given as well.