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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
DOI: https://doi.org/10.1090/S0002-9947-03-03355-5
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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  • 1. R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics 169, Springer-Verlag, New York, 1997. MR 98i:15003
  • 2. W. Donoghue, Monotone matrix functions and analytic continuation, Grundlehren der mathematischen Wissenschaften 207, Springer-Verlag, New York and Heidelberg, 1974. MR 58:6279
  • 3. P. Borwein and T. Erdelyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics 161, Springer-Verlag, New York, 1995. MR 97e:41001
  • 4. F. Hansen and G. K. Pedersen, Jensen's inequality for operators and Löwner's theorem, Math. Ann. 258 (1982), 229-241. MR 83g:47020
  • 5. R. Horn and C. Johnson, Topics in matrix analysis, Cambridge Univ. Press, 1991. MR 92e:15003
  • 6. A. Koranyi, On a theorem of Löwner and its connections with resolvents of selfadjoint transformations, Acta Sci. Math. Szeged 17 (1956), 63-70. MR 18:588c
  • 7. K. Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177-216.
  • 8. M. Rosenblum and J. Rovnyak, Hardy classes and operator theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, 1985. MR 87e:47001
  • 9. M. Uchiyama, Operator monotone functions which are defined implicitly and operator inequalities, J. Funct. Anal. 175 (2000), 330-347. MR 2001h:47021
  • 10. M. Uchiyama and M. Hasumi, On some operator monotone functions, Integral Equations Operator Theory 42 (2002), 243-251. MR 2002k:47044

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Additional Information

Mitsuru Uchiyama
Affiliation: Department of Mathematics, Fukuoka University of Education, Munakata, Fukuoka, 811-4192, Japan
Email: uchiyama@fukuoka-edu.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-03-03355-5
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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