Inverse functions of polynomials and orthogonal polynomials as operator monotone functions

• Published in: Transactions of the American Mathematical Society Vol. 355; no. 10; pp. 4111 - 4123
• Main Author: Uchiyama, Mitsuru
• Format: Journal Article
• Language: English
• Published: Providence, RI American Mathematical Society 01-10-2003
• Subjects: Algebra; Coefficients; Continuous functions; Exact sciences and technology; Functions of a complex variable; Inverse functions; Linear and multilinear algebra, matrix theory; Mathematical analysis; Mathematical functions; Mathematical inequalities; Mathematical monotonicity; Mathematical theorems; Mathematics; Monotonic functions; Operator theory; Polynomials; Real numbers; Sciences and techniques of general use; Special functions; Positive semi-definite operator; operator monotone function; orthogonal polynomials; Herglotz theorem; Polynomial inverse function; Operator function; Hilbert space; Monotone operator; Monotonicity; Legendre polynomial; Orthogonal polynomial; Operator theory; Convergence; Inverse problem
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-03-03355-5

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.