# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Operator monotone functions, positive definite kernels and majorizationHTML articles powered by AMS MathViewer

by Mitsuru Uchiyama
Proc. Amer. Math. Soc. 138 (2010), 3985-3996 Request permission

## 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.
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