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

TL;DR

This paper characterizes all linear operators that preserve elliptic, positive, and non-negative polynomials in one and higher dimensions, using advanced mathematical tools, and addresses longstanding questions related to entire functions and Hilbert's 17th problem.

Contribution

It provides a comprehensive classification of such operators in finite and infinite-dimensional polynomial spaces, extending previous results and solving open problems.

Findings

Complete characterization of operators preserving polynomial positivity

Extension of results to higher-dimensional polynomial spaces

Resolution of questions related to entire functions and Hilbert's 17th problem

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.