Weighted means and Karcher equations of positive operators

The Karcher or least-squares mean has recently become an important tool for the averaging and study of positive definite matrices. In this paper, we show that this mean extends, in its generalweighted 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 Pt 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+. We show each of these characterizations provide important insights about the Karcher mean.