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Operator monotone functions, positive definite kernels and majorization

Author: Mitsuru Uchiyama
Journal: Proc. Amer. Math. Soc. 138 (2010), 3985-3996
MSC (2010): Primary 47A56; Secondary 15A39, 47B34
Published electronically: May 10, 2010
MathSciNet review: 2679620
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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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Additional Information

Mitsuru Uchiyama
Affiliation: Department of Mathematics, Interdisciplinary Faculty of Science and Engineering, Shimane University, Matsue City, Shimane 690-8504, Japan

Keywords: Positive definite kernel, pick function, matrix order, L\"{o}wner theorem, operator monotone function, majorization, infinitely divisible kernel
Received by editor(s): September 1, 2009
Received by editor(s) in revised form: October 2, 2009, December 14, 2009, and January 17, 2010
Published electronically: May 10, 2010
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society

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