## 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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