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 $K\subseteq\mathbb{R}^n$, $e^A\mathrm{Pos}(K)_{\leq 2d}\subseteq\sum\mathbb{R}[x_1,\dots,x_n]_{\leq d}^2$. We give an explicit operator $A$ such that $e^A\mathrm{Pos}(\mathbb{R}^n)\subseteq\sum\mathbb{R}[x_1,\dots,x_n]^2$. For $K\subseteq\mathbb{R}^n$, we give a condition such that $A$ exists with $e^A\mathrm{Pos}(K)\subseteq\sum\mathbb{R}[x_1,\dots,x_n]^2$. 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 $T\mathrm{Pos}(K)\subseteq\sum\mathbb{R}[x_1,\dots,x_n]^2$. In the framework of regular Fr\'echet 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.