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 ${\left(B^{ts/2}{A}^{(1-t)s}{B}^{ts/2}\right)}^{1/s}$ 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.

20 pages; 2 figures