Inequalities and limits of weighted spectral geometric mean

TL;DR

This paper investigates new properties and inequalities of the spectral geometric mean of positive semidefinite matrices, including log majorization relations and limits, with extensions to symmetric spaces of negative curvature.

Contribution

It introduces novel inequalities and limit relations for the spectral geometric mean, extending results to symmetric spaces of negative curvature.

Findings

Proved a log majorization relation involving spectral geometric mean.

Analyzed the limit behavior of the $t$-spectral mean.

Extended results to symmetric spaces of negative curvature.

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 $A\natural_t B :=(A^{-1}\sharp B)^{t}A(A^{-1}\sharp B)^{t}$ of two positive semidefinite matrices $A$ and $B$, where $A\sharp B$ 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.