On linear operators preserving the set of positive polynomials

Following the classical approach of Pólya-Schur theory [13] we initiate in this paper the study of linear operators acting on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}[x]$$\end{document} and preserving either the set of positive univariate polynomials or similar sets of non-negative and elliptic polynomials.