Classifications of linear operators preserving elliptic, positive and non-negative polynomials
Classical Analysis and ODEs
2009-02-04 v3 Functional Analysis
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 [9] originating from entire function theory and the literature pertaining to Hilbert's 17th problem.
Cite
@article{arxiv.0811.4374,
title = {Classifications of linear operators preserving elliptic, positive and non-negative polynomials},
author = {Julius Borcea},
journal= {arXiv preprint arXiv:0811.4374},
year = {2009}
}
Comments
Final version, to appear in J. Reine Angew. Math. (Crelle); 12 pages, no figures, LaTeX2e