On Positivity Preservers with constant Coefficients and their Generators

Abstract

In this work we study positivity preservers $T:\mathbb{R}[x_1,\dots ,x_n]\to \mathbb{R}[x_1,\dots ,x_n]$ with constant coefficients and define their generators $A$ if they exist, i.e., $\exp (A)=T$. We use the theory of regular Fréchet Lie groups to show the first main result. A positivity preserver with constant coefficients has a generator if and only if it is represented by an infinitely divisible measure (Main Theorem 4.7). In the second main result (Main Theorem 4.11) we use the Lévy--Khinchin formula to fully characterize the generators of positivity preservers with constant coefficients.

Following external advice we included more detailed Lie group arguments to prove the two main theorems