Operator monotone functions, positive definite kernels and majorization

Abstract:

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 $F(t)=\int f(t)dt$ the operator function $F(X)$ is convex. Let $h(t)$ and $g(t)$ be $C^1$ 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 $K_{h,\;g}$ by $K_{h,\;g}(t,s)=(h(t)-h(s))/(g(t)-g(s))$, which is a generalization of the Löwner kernel function. We will see that it is positive definite if and only if $h(A)\leqq h(B)$ whenever $g(A)\leqq g(B)$ for Hermitian operators $A, B$, and we give a method to construct a large number of infinitely divisible kernel functions.