Operator Spectral Geometric Versus Geometric Mean

TL;DR

This paper introduces new inequalities for the spectral geometric mean of positive definite operators, comparing it with the geometric mean and exploring implications for related inequalities in operator theory.

Contribution

It presents novel inequalities for the spectral geometric mean and explicit comparisons with the geometric mean, extending existing operator inequalities.

Findings

New inequalities for spectral geometric mean

Explicit comparison between spectral and geometric means

Applications to Ando-type and Ando-Hiai inequalities

Abstract

The main goal of this article is to present new inequalities for the spectral geometric mean $A\natural_t B$ of two positive definite operators $A, B$ on a Hilbert space. The obtained results complement many known inequalities for the geometric mean $A\sharp_t B$. In particular, explicit comparisons between $A\natural_t B$ and $A\sharp_t B$ are given, with applications towards Ando-type inequalities and Ando-Hiai inequalities for $A\natural_t B$ and some other consequences.