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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let be a sequence of orthonormal polynomials and the restriction of to , where is the maximum zero of . Then and the composite are operator monotone on . Furthermore, for every polynomial with a positive leading coefficient there is a real number so that the inverse function of defined on is semi-operator monotone, that is, for matrices , implies

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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