Operator means, barycenters, and fixed point equations

TL;DR

This paper surveys the intersection of algebraic and geometric methods in defining and understanding operator means, barycenters, and fixed point equations for positive operators.

Contribution

It highlights cases where algebraic and geometric approaches to operator means converge, providing a unified perspective on their characterization.

Findings

Identification of conditions where algebraic and geometric methods agree

Characterization of operator means via fixed point equations

Insights into the structure of barycenters in positive operators

Abstract

The seminal work of Kubo and Ando from 1980 provided us with an axiomatic approach to means of positive operators. As most of their axioms are algebraic in nature, this approach has a clear algebraic flavor. On the other hand, it is highly natural to take the geometric viewpoint and consider a distance (understood in a broad sense) on the cone of positive operators, and define the mean of positive operators by an appropriate notion of the center of mass. This strategy often leads to a fixed point equation that characterizes the mean. The aim of this survey is to highlight those cases where the algebraic and the geometric approaches meet each other.