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

TL;DR

This paper characterizes diagonal positivity preservers and their generators, explores their relation to moment sequences, and provides new proofs and representations that simplify existing literature on positivity-preserving linear maps.

Contribution

It offers a full characterization of diagonal positivity preservers, their generators, and connects these to infinitely divisible moment sequences, advancing the understanding of positivity-preserving linear maps.

Findings

Full characterization of diagonal positivity preservers

Representation of such maps and their generators

Connection to infinitely divisible moment sequences

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 $Tx^\alpha = t_\alpha x^\alpha$ holds for all $\alpha\in\mathbb{N}_0^n$ with $t_\alpha\in\mathbb{R}$ and $Tp\geq 0$ on $\mathbb{R}^n$ for all $p\in\mathbb{R}[x_1,\dots,x_n]$ with $p\geq 0$ 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 $t\geq 0$. We give the connection of these generators to infinitely divisible moment sequences.