TY - JOUR

T1 - Convex multivariate operator means

AU - Hansen, Frank

N1 - Publisher Copyright:
© 2018 Elsevier Inc.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/3/1

Y1 - 2019/3/1

N2 - 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.

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

UR - http://www.scopus.com/inward/record.url?scp=85058110769&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85058110769&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2018.11.032

DO - 10.1016/j.laa.2018.11.032

M3 - Article

AN - SCOPUS:85058110769

VL - 564

SP - 209

EP - 224

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -