The Hadamard Product of Moment Sequences, Diagonal Positivity Preservers, and their Generators
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866929604960190464 |
|---|---|
| author | di Dio, Philipp J. Langer, Lars-Luca |
| author_facet | di Dio, Philipp J. Langer, Lars-Luca |
| contents | 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 $Tx^α= t_αx^α$ holds for all $α\in\mathbb{N}_0^n$ with $t_α\in\mathbb{R}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19933 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Hadamard Product of Moment Sequences, Diagonal Positivity Preservers, and their Generators di Dio, Philipp J. Langer, Lars-Luca Functional Analysis Algebraic Geometry 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 $Tx^α= t_αx^α$ holds for all $α\in\mathbb{N}_0^n$ with $t_α\in\mathbb{R}$ 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. |
| title | The Hadamard Product of Moment Sequences, Diagonal Positivity Preservers, and their Generators |
| topic | Functional Analysis Algebraic Geometry |
| url | https://arxiv.org/abs/2407.19933 |