The Pólya-Schur Problem on the Unit Circle

C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (4) 2015, pp. 131-141
(Received: 2014-10-06 , Revised: 2014-10-24)

Peter C. Gibson,Dept. of Mathematics & Statistics, York University, 4700 Keele St., Toronto, Ontario,
Canada, M3J 1P3; e-mail: pcgibson@yorku.ca

Abstract/Résumé:

The Pólya-Schur problem for a region \(Z\) in the complex plane is to characterize the semigroup of linear operators \(A:\mathbb{C}[z]\rightarrow \mathbb{C}[z]\) that map polynomials whose zeros are confined to \(Z\) to polynomials of the same type, or to 0. We give a constructive solution to the Pólya-Schur problem in the case where \(Z\) is the unit circle. This shows that the associated semigroup is qualitatively simpler than in the classical case where \(Z\) is the real line, whereas recent results have not clearly distinguished the two cases.

Le problème Pólya-Schur pour une région \(Z\) dans le plan complexe est de charactériser le semigroupe des opérateurs linéaires \(A:\mathbb{C}[z]\rightarrow \mathbb{C}[z]\) envoyant chaque polynôme dont les racines appartiennent à \(Z\) vers un polynôme du même type, ou vers 0. Nous présentons une solution constructive au problème Pólya-Schur dans le cas où \(Z\) est le cercle unité. Cela démontre que le semigroupe associé est qualitativement plus simple que dans le cas classique de la ligne réelle, tandis que les résultats récents n’ont pas distingué les deux cas.

Keywords: Polya-Schur type theorems, composition operators, stable polynomials

AMS Subject Classification: Polynomials, Zeros of polynomials; rational functions; and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral), Operators on function spaces (general) 30C10, 30C15, 47B38

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