A fixed point mean approximation to the Cartan barycenter of positive definite matrices

Kim, Sejong; Lee, Hosoo; Lim, Yongdo

doi:10.1016/j.laa.2016.02.005

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Cited by 4 publications

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“…In recent study of operator means, the fixed point method, apart from the Riemannian geometry method, provides a main technical tool as used in different places in, e.g., [24,26,30,31,34,35,36,40,48]. In particular, the Karcher and the power means are defined as the solutions to certain fixed point type equations.…”

Section: Introductionmentioning

confidence: 99%

Operator means of probability measures

Hiai

Lim

2020

Advances in Mathematics

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Let P be the complete metric space consisting of positive invertible operators on an infinite-dimensional Hilbert space with the Thompson metric. We introduce the notion of operator means of probability measures on P, in parallel with Kubo and Ando's definition of two-variable operator means, and show that every operator mean is contractive for the ∞-Wasserstein distance. By means of a fixed point method we consider deformation of such operator means, and show that the deformation of any operator mean becomes again an operator mean in our sense. Based on this deformation procedure we prove a number of properties and inequalities for operator means of probability measures.

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