The Hadamard Product of Moment Sequences, Diagonal Positivity Preservers, and their Generators

Abstract

In this work we investigate special aspects of positivity preservers and especially diagonal positivity preservers, i.e., linear maps $T:\mathbb{R}[x_1,\dots ,x_n]\to \mathbb{R}[x_1,\dots ,x_n]$ such that $T{x}^{\alpha }=t_{\alpha }{x}^{\alpha }$ holds for all $\alpha \in {\mathbb{N}}_0^n$ with $t_{\alpha }\in \mathbb{R}$ and $Tp\ge 0$ on ${\mathbb{R}}^n$ for all $p\in \mathbb{R}[x_1,\dots ,x_n]$ with $p\ge 0$ on ${\mathbb{R}}^n$. We discuss representations of $T$, give characterizations of diagonal positivity preservers, and compare these to previous (partial) results in the literature. On the side we get a full characterization of linear maps preserving moment sequences and a new proof of Schur's product formula. The tool of diagonal positivity preservers simplifies several other existing proofs in the literature. We give a full characterization of generators $A$ of diagonal positivity preservers, i.e., $e^{tA}$ is a diagonal positivity preserver for all $t\ge 0$. We give the connection of these generators to infinitely divisible moment sequences.