Operator inequalities and gyrolines of the weighted geometric means

TL;DR

This paper explores geometric means of positive definite operators, establishing their properties, inequalities, and connections to gyrogroups, with explicit formulas for 2x2 matrices and density matrices.

Contribution

It introduces a bijection between different weighted geometric means and reveals their geometric and algebraic properties within gyrogroup structures.

Findings

Component-wise bijection of geometric means

Spectral geometric mean as a metric midpoint

Explicit formulas for 2x2 positive definite matrices

Abstract

We consider in this paper two different types of the weighted geometric means of positive definite operators. We show the component-wise bijection of these geometric means and give a geometric property of the spectral geometric mean as a metric midpoint. Moreover, several interesting inequalities related with the geometric means of positive definite operators will be shown. We also see the meaning of weighted geometric means in the gyrogroup structure with finite dimension and find the formulas of weighted geometric means of 2-by-2 positive definite matrices and density matrices.