Title

Weighted means and Karcher equations of positive operators

Document Type

Article

Publication Date

9-24-2013

Abstract

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.

Publication Source (Journal or Book title)

Proceedings of the National Academy of Sciences of the United States of America

First Page

15626

Last Page

15632

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