Revisit on spectral geometric mean

TL;DR

This paper explores the spectral geometric mean of positive definite operators, establishing its properties as a geodesic, its relation to tolerance relations, and its ability to pinch positive tuples, thus deepening understanding of spectral means.

Contribution

It introduces the limit and unique solution of nonlinear equations related to the spectral geometric mean, and characterizes tolerance relations and pinching properties.

Findings

Spectral geometric mean is a geodesic with respect to a semi-metric.

Tolerance relations on determinant one matrices are characterized by the spectral geometric mean.

Spectral geometric mean can pinch two positive tuples.

Abstract

In this paper we introduce the limit, unique solution of the nonlinear equations, geodesic property, tolerance relations and pinch on the spectral geometric mean for two positive definite operators. We show that the spectral geometric mean is a geodesic with respect to some semi-metric. We also prove that the tolerance relation on determinant one matrices can be characterized by the spectral geometric mean. Moreover, two positive tuples can be pinched by the spectral geometric mean.