On the Ando-Hiai property for spectral geometric means

TL;DR

This paper investigates properties of spectral geometric means of positive definite matrices, establishing fundamental operator function properties, inequalities, and log-majorization relations, advancing understanding in matrix analysis.

Contribution

It introduces a two-variable operator function including spectral geometric means and proves the Ando-Hiai inequality under specific conditions, providing new insights.

Findings

Proves fundamental properties of the operator function involving spectral geometric means.

Establishes the Ando-Hiai type inequality for spectral geometric means.

Derives log-majorization relations and norm inequalities for positive definite matrices.

Abstract

In this paper, we consider a two-variable operator function that includes two weighted spectral geometric means, and show fundamental properties of the operator function. Moreover, it satisfies the Ando-Hiai type inequality under some restricted conditions. As an application, we show the log-majorization relations and norm inequalities for the spectral geometric means of positive definite matrices.