$K$-Positivity Preservers and their Generators

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Abstract

We study $K$-positivity preservers with given closed $K\subseteq\mathbb{R}^n$, i.e., linear maps $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ such that $T\mathrm{Pos}(K)\subseteq\mathrm{Pos}(K)$ holds, and their generators $A:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$, i.e., $e^{tA}\mathrm{Pos}(K)\subseteq\mathrm{Pos}(K)$ holds for all $t\geq 0$. We characterize these maps $T$ for any closed $K\subseteq\mathbb{R}^n$ in Theorem 4.5. We characterize the maps $A$ in Theorem 5.12 for $K=\mathrm{R}^n$ and give partial results for general $K$. In Proposition 6.1 and 6.3 we give maps $A$ such that $e^{tA}$ is a positivity preserver for all $t\geq \tau$ for some $\tau>0$ but not for $t\in (0,\tau)$, i.e., we have an eventually positive semi-group.