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 the continuous kernel function Kh, g by Kh, g(t, s) = (h(t) − h(s))/(g(t) − g(s)), which is a generalization of the Löwner kernel function. We will see that it is positive definite if and only if h(A) h(B) whenever g(A) g(B) for Hermitian operators A,B, and we give a method to construct a large number of infinitely divisible kernel functions.