On infinite divisibility of positive definite functions arising from operator means

Operator Monotone Functions, Positive Definite Kernels and Majorization

Let f(t) be a real continuous function on an interval, and consider the operator function f(X) defined for Hermitian operators X. We will show that if f(X) is increasing w.r.t. the operator order, then for F (t) = ∫ f(t)dt the operator function F (X) is convex. Let h(t) and g(t) be C1 functions defined on an interval I. Suppose h(t) is non-decreasing and g(t) is increasing. Then we will define ...

Operator norms of words formed from positive-definite matrices

Let α1, α2, . . . , αn, β1, β2, . . . , βn be strictly positive reals with α1 +α2 + · · ·+αn = β1 + β2 + · · ·+ βn = s. In this paper, the inequality |||Aα1Bβ1Aα2 · · ·AαnBβn ||| ≤ |||AB|||s when A and B are positive-definite matrices is studied. Related questions are also studied.

GENERALIZED POSITIVE DEFINITE FUNCTIONS AND COMPLETELY MONOTONE FUNCTIONS ON FOUNDATION SEMIGROUPS

A general notion of completely monotone functionals on an ordered Banach algebra B into a proper H*-algebra A with an integral representation for such functionals is given. As an application of this result we have obtained a characterization for the generalized completely continuous monotone functions on weighted foundation semigroups. A generalized version of Bochner’s theorem on foundation se...

Multivariate positive definite functions on spheres

In 1942 I.J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a positive linear combination of the Gegenbauer polynomials. In this paper we extend Schoenberg’s theorem for multivariate Gegenbauer polynomials. This extension derives new positive semidefinite constraints for the distance distribution which can be applied for spherical codes.

Schoenberg’s Problem on Positive Definite Functions