Classifications of linear operators preserving elliptic, positive and non-negative polynomials : Journal für die reine und angewandte Mathematik (Crelles Journal)

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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk


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Classifications of linear operators preserving elliptic, positive and non-negative polynomials

1Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2011, Issue 650, Pages 67–82, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle.2011.003, January 2011

Publication History

Received:
2008-10-17
Revised:
2009-01-12
Published Online:
2011-01-07

Abstract

We characterize all linear operators on finite or infinite-dimensional spaces of univariate real polynomials preserving the sets of elliptic, positive, and non-negative polynomials, respectively. This is done by means of Fischer–Fock dualities, Hankel forms, and convolutions with non-negative measures. We also establish higher-dimensional analogs of these results. In particular, our classification theorems solve the questions raised in [Borcea, Guterman, Shapiro, Preserving positive polynomials and beyond] originating from entire function theory and the literature pertaining to Hilbert's 17th problem.

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