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Journal of Operator Theory

Volume 77, Issue 1, Winter 2017  pp. 87-107.

Contractive barycentric map

Authors:  Jimmie D. Lawson (1) and Yongdo Lim (2)
Author institution:(1) Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A.
(2) Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea

Summary: We first develop in the context of complete metric spaces a one-to-one correspondence between the class of means $G=\{G_n\}_{n\geqslant 2}$ that are symmetric, multiplicative, and contractive and the class of contractive (with respect to the Wasserstein metric) barycentric maps on the space of $L^1$-prob-ability measures. We apply this equivalence to the recently introduced and studied Karcher mean on the open cone $\mathbb{P}$ of positive invertible operators on a Hilbert space equipped with the Thompson metric to obtain a corresponding contractive barycentric map. In this context we derive a version of earlier results of Sturm and Lim and Palfia about approximating the Karcher mean with the more constructive inductive mean. This leads to the conclusion that the Karcher barycenter lies in the strong closure of the convex hull of the support of a probability measure. This fact is a crucial ingredient in deriving a version of Jensen's inequality, with which we close.

Keywords: positive operator, operator mean, barycentric map, Karcher mean, geodesic metric space, Jensen's inequality

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