In-betweenness: a geometric monotonicity property for operator means

TL;DR

This paper introduces the concepts of in-betweenness and monotonicity for operator means, generalizing scalar mean properties and exploring their behavior under different metrics, including Euclidean and trace metrics.

Contribution

It defines new geometric properties for operator means and demonstrates monotonicity for specific classes and all Kubo-Ando means under relevant metrics.

Findings

Two classes of non-trivial means are monotonic with the Euclidean metric.

All Kubo-Ando means are monotonic with respect to the trace metric.

The notions generalize scalar mean properties and relax geodesity.

Abstract

We introduce the notions of in-betweenness and monotonicity with respect to a metric, for operator means. These notions can be seen as generalising their natural counterpart for scalar means, and as a relaxation of the notion of geodesity. We exhibit two classes of non-trivial means that are monotonic with respect to the Euclidean metric. We also show that all Kubo-Ando means are monotonic with respect to the trace metric, which is the natural metric for the geometric mean.