The ALM-Framework in the Theory of Multivariate Operator Means

TL;DR

This paper generalizes the ALM-procedure for multivariate operator means, proving it preserves key properties and exploring the self-adjointness of the resulting geometric mean, advancing the theoretical framework of operator means.

Contribution

It introduces a generalized ALM-procedure for multivariate operator means, ensuring property preservation and analyzing self-adjointness, thus extending the theoretical understanding of operator means.

Findings

The generalized ALM-procedure preserves all axiomatic properties of operator means.

The procedure maintains several additional desirable properties.

The self-adjointness of the constructed geometric mean is examined.

Abstract

In this paper, we generalize the ALM-procedure introduced by Ando, Li, and Mathias for extending operator geometric means to multiple variables. We prove that the generalized procedure preserves all the properties required by the axioms of operator means, along with several additional desirable properties. Moreover, we examine the self-adjointness of the canonical geometric mean constructed through this procedure.