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Inverse functions of polynomials and orthogonal polynomials as operator monotone functions

Author: Mitsuru Uchiyama
Journal: Trans. Amer. Math. Soc. 355 (2003), 4111-4123
MSC (2000): Primary 47A63, 15A48; Secondary 33C45, 30B40
Published electronically: June 10, 2003
MathSciNet review: 1990577
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Abstract: We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let $\{p_n\}_{n=0}^{\infty}$ be a sequence of orthonormal polynomials and $p_{n+}$ the restriction of $p_n$ to $[a_n, \infty)$, where $a_n$ is the maximum zero of $p_n$. Then $p_{n+}^{-1}$ and the composite $p_{n-1}\circ p_{n+}^{-1}$ are operator monotone on $[0, \infty)$. Furthermore, for every polynomial $p$ with a positive leading coefficient there is a real number $a$ so that the inverse function of $p(t+a)-p(a)$ defined on $[0,\infty)$is semi-operator monotone, that is, for matrices $ A,B \geq 0$, $(p(A+a)-p(a))^2 \leq ((p(B+a)-p(a))^{2}$ implies $A^2\leq B^2.$

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Additional Information

Mitsuru Uchiyama
Affiliation: Department of Mathematics, Fukuoka University of Education, Munakata, Fukuoka, 811-4192, Japan

Keywords: Positive semi-definite operator, operator monotone function, orthogonal polynomials
Received by editor(s): October 16, 2002
Published electronically: June 10, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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