Operator Spectral Geometric Versus Geometric Mean

Abstract

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

arXiv commentThe coefficients in Theorem 2.2 and Corollary 2.3 have been changed. To appear in LAMA