Making Non-Negative Polynomials into Sums of Squares

We investigate linear operators $A:\mathbb{R}[x_1,\dots ,x_n]\to \mathbb{R}[x_1,\dots ,x_n]$. We give explicit operators $A$ such that, for fixed $d\in {\mathbb{N}}_0$ and closed $K\subseteq {\mathbb{R}}^n$, $e^A\mathrm{P}\mathrm{o}\mathrm{s}(K{)}_{\le 2d}\subseteq \sum \mathbb{R}[x_1,\dots ,x_n{]}_{\le d}^2$. We give an explicit operator $A$ such that $e^A\mathrm{P}\mathrm{o}\mathrm{s}({\mathbb{R}}^n)\subseteq \sum \mathbb{R}[x_1,\dots ,x_n{]}^2$. For $K\subseteq {\mathbb{R}}^n$, we give a condition such that $A$ exists with $e^A\mathrm{P}\mathrm{o}\mathrm{s}(K)\subseteq \sum \mathbb{R}[x_1,\dots ,x_n{]}^2$. We show that, for compact $K\subseteq {\mathbb{R}}^n$, there is no bijective linear operator $T:\mathbb{R}[x_1,\dots ,x_n]\to \mathbb{R}[x_1,\dots ,x_n]$ with $T\mathrm{P}\mathrm{o}\mathrm{s}(K)\subseteq \sum \mathbb{R}[x_1,\dots ,x_n{]}^2$. In the framework of regular Fréchet Lie groups and Lie algebras we investigate the linear operators $A$ such that $e^{tA}:\mathbb{R}[x_1,\dots ,x_n]\to \mathbb{R}[x_1,\dots ,x_n]$ is well-defined for all $t\in \mathbb{R}$. We give a three-line-proof of Stochel’s Theorem.