Karcher means and Karcher equations of positive definite operators

Abstract

The Karcher or least-squares mean has recently become an important tool for the averaging and studying of positive definite matrices. In this paper we show that this mean extends, in its general weighted form, to the infinite-dimensional setting of positive operators on a Hilbert space and retains most of its attractive properties. The primary extension is via its characterization as the unique solution of the corresponding Karcher equation. We also introduce power means P t P_t in the infinite-dimensional setting and show that the Karcher mean is the strong limit of the monotonically decreasing family of power means as t → 0 + t\to 0^+ . We show that each of these characterizations provide important insights about the Karcher mean.