On Positivity Preservers with constant Coefficients and their Generators

TL;DR

This paper characterizes positivity preservers with constant coefficients and their generators using Lie group theory and the Lévy–Khinchin formula, establishing conditions for their existence and explicit descriptions.

Contribution

It provides a complete characterization of generators for positivity preservers with constant coefficients via measure theory and Lie group methods.

Findings

A positivity preserver has a generator iff it is represented by an infinitely divisible measure.

The generators are fully characterized using the Lévy–Khinchin formula.

The study connects positivity preservers with advanced measure and Lie group theories.

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 Fr\'echet 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 L\'evy--Khinchin formula to fully characterize the generators of positivity preservers with constant coefficients.