Convex multivariate operator means

Frank Hansen

doi:10.1016/j.laa.2018.11.032

Convex multivariate operator means

Frank Hansen

Division of Developmental Research in Education Programs

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

We prove that a trace function, generated by the functional calculus, is geodesically convex in the Riemannian manifold of positive definite matrices, if and only if it is geodesically convex in positive numbers. The analysis of multivariate operator means is facilitated by the study of classes of means that are fix-points under a contraction with respect to the Thompson metric. Although such methods are powerful, they crucially depend on monotonicity. We develop new techniques to prove existence of multivariate operator means that are not necessarily monotone.

Original language	English
Pages (from-to)	209-224
Number of pages	16
Journal	Linear Algebra and Its Applications
Volume	564
DOIs	https://doi.org/10.1016/j.laa.2018.11.032
Publication status	Published - 2019 Mar 1

Keywords

Convex-log function
Geodesically convex function
Hyper-mean
Multivariate operator mean
Operator mean

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/j.laa.2018.11.032

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AB - We prove that a trace function, generated by the functional calculus, is geodesically convex in the Riemannian manifold of positive definite matrices, if and only if it is geodesically convex in positive numbers. The analysis of multivariate operator means is facilitated by the study of classes of means that are fix-points under a contraction with respect to the Thompson metric. Although such methods are powerful, they crucially depend on monotonicity. We develop new techniques to prove existence of multivariate operator means that are not necessarily monotone.

KW - Convex-log function

KW - Geodesically convex function

KW - Hyper-mean

KW - Multivariate operator mean

KW - Operator mean

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JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -

Convex multivariate operator means

Abstract

Keywords

ASJC Scopus subject areas

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