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

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.