On Positivity Preservers and their Generators

We study \(K\)-positivity preserves \(T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]\) with constant coefficients and \(K\subseteq\mathbb{R}^n\); especially their generators \(A\), i.e., \(T = e^A\). We completely describe the set of generators \(A\) such that \(e^{tA}\) is a positivity pre...

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Bibliographic Details
Published in:arXiv.org 2023-08
Main Author: di Dio, Philipp J
Format: Article
Language:English
Subjects:
Functions (mathematics)
Group theory
Lie groups
Mathematical analysis
Polynomials
Theorems
Online Access:Get full text
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Summary:We study \(K\)-positivity preserves \(T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]\) with constant coefficients and \(K\subseteq\mathbb{R}^n\); especially their generators \(A\), i.e., \(T = e^A\). We completely describe the set of generators \(A\) such that \(e^{tA}\) is a positivity preserver for all \(t\geq 0\) (Theorem 4.8). We solve the long standing open problem of describing the \(K\)-positivity preservers with constant coefficients when \(K\subsetneq\mathbb{R}^n\) (Theorem 3.4). The complete description of generators of \([0,\infty)\)-positivity preservers is given in Theorem 6.4. We give an example of a strange positivity action which on \([0,\infty)\) maps odd non-negative polynomials to odd non-negative polynomials but even non-negative polynomials are mapped to non-negative functions which are not polynomials. We introduce the concept of a convolution of (moment) sequences. The techniques we use come from (infinite dimensional matrix) Lie group theory, measure theory (infinitely divisible measures and the Lévy--Khinchin formula), and of course the theory of moments.
ISSN:2331-8422