TY - JOUR
T1 - Convex multivariate operator means
AU - Hansen, Frank
PY - 2019
Y1 - 2019
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
U2 - 10.1016/j.laa.2018.11.032
DO - 10.1016/j.laa.2018.11.032
M3 - Journal 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 -