Convex multivariate operator means

TL;DR

This paper introduces a novel technique for establishing the existence of non-monotonic multivariate operator means, expanding the scope beyond traditional monotone methods based on fixed points and the Thompson metric.

Contribution

It develops a new approach to prove the existence of multivariate operator means that do not rely on monotonicity, creating a new class of such means.

Findings

Established a method for non-monotonic multivariate operator means

Expanded the theoretical framework beyond Thompson metric-based fixed points

Introduced new classes of operator means with potential applications

Abstract

The dominant method for defining multivariate operator means is to express them as fix-points under a contraction with respect to the Thompson metric. Although this method is powerful, it crucially depends on monotonicity. We are developing a technique to prove the existence of multivariate operator means that are not necessarily monotone. This gives rise to an entire new class of non-monotonic multivariate operator means.