Inequalities and limits of weighted spectral geometric mean

Abstract

We establish some new properties of spectral geometric mean. In particular, we prove a log majorization relation between B ts/2 A (1−t)s B ts/2 1/s and the t-spectral mean A t B := (A −1 B) t A(A −1 B) t of two positive semidefinite matrices A and B, where A B is the geometric mean, and the t-spectral mean is the dominant one. The limit involving t-spectral mean is also studied. We then extend all the results in the context of symmetric spaces of negative curvature.