2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/218355We 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, LaTeX2eClassical Analysis and ODEsFunctional Analysis12D10, 12D15, 15A04, 30C15, 32A60, 46E22, 47B38text