In this work we investigate special aspects of positivity preservers and especially diagonal positivity preservers, i.e., linear maps \(T:\mathbb {R}[x_1,\dots ,x_n]\rightarrow \mathbb {R}[x_1,\dots ,x_n]\) such that \(Tx^\alpha = t_\alpha x^\alpha \) 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 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.
