Classifications of linear operators preserving elliptic, positive and non-negative polynomials
arXiv:0811.4374
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.
Final version, to appear in J. Reine Angew. Math. (Crelle); 12 pages, no figures, LaTeX2e