Semigroups of operator means and generalized Karcher equations

TL;DR

This paper extends the theory of operator means on Hilbert spaces using differential geometry, establishing unique solutions to generalized Karcher equations via semigroups of holomorphic functions and classifying when these solutions form geodesics.

Contribution

It introduces a geometric framework for operator means, generalizes Karcher equations, and characterizes when these equations' solutions are geodesics in the manifold of positive operators.

Findings

Unique solutions to generalized Karcher equations are obtained as limits of induced operator means.

Semigroups of holomorphic functions induce operator means that behave as geodesics under certain conditions.

The paper classifies cases where these operator means are geodesics of affinely connected manifolds.

Abstract

In this article we consider means of positive operators on a Hilbert space. We extend the theory of matrix power means to arbitrary operator means in the sense of Kubo-Ando. The basis of the extension is relying on ideas coming from differential geometry. We consider generalized Karcher equations for positive operators and show that such equations admit unique positive solutions that can be obtained as a limit of one-parameter families of operator means called induced operator means. These means are themselves unique fixed points of one parameter families of strict contractions induced, through Kubo-Ando theory of operator means, by semigroups of holomorphic functions mapping the upper half-plane into itself. These semigroups of holomorphic functions are considered with Koenigs function corresponding to Schroeder's functional equation. Koenigs function in this setting provides us with a logarithm map corresponding to every 2-variable operator mean. The semigroups of 2-variable means behave as geodesics and we exactly classify the cases when they are indeed geodesics of affinely connected manifolds, thereby providing the cases when these generalized Karcher equations are exactly Karcher equations in the geometric sense. This is achieved by studying the arising holonomy groups. The unique solutions of these generalized Karcher equations are called lambda extensions and have numerous desirable properties which are inherited from the induced operator means themselves.