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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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 majorization
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by Mitsuru Uchiyama PDF
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.
References
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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
  • MR Author ID: 198919
  • Email: uchiyama@riko.shimane-u.ac.jp
  • 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
  • © Copyright 2010 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 138 (2010), 3985-3996
  • MSC (2010): Primary 47A56; Secondary 15A39, 47B34
  • DOI: https://doi.org/10.1090/S0002-9939-10-10386-4
  • MathSciNet review: 2679620