On Positivity Preservers with constant Coefficients and their Generators
Abstract
In this work we study positivity preservers $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ with constant coefficients and define their generators $A$ if they exist, i.e., $\exp(A) = T$. We use the theory of regular Fréchet Lie groups to show the first main result. A positivity preserver with constant coefficients has a generator if and only if it is represented by an infinitely divisible measure (Main Theorem 4.7). In the second main result (Main Theorem 4.11) we use the Lévy--Khinchin formula to fully characterize the generators of positivity preservers with constant coefficients.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2023
- DOI:
- 10.48550/arXiv.2308.10455
- arXiv:
- arXiv:2308.10455
- Bibcode:
- 2023arXiv230810455D
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Functional Analysis;
- Primary 44A60;
- 47A57;
- 15A04;
- Secondary 12D15;
- 45P05;
- 47B38
- E-Print:
- Following external advice we included more detailed Lie group arguments to prove the two main theorems