On Positivity Preservers with constant Coefficients and their Generators
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913338499268608 |
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| author | di Dio, Philipp J. |
| author_facet | di Dio, Philipp J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_10455 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Positivity Preservers with constant Coefficients and their Generators di Dio, Philipp J. Algebraic Geometry Functional Analysis Primary 44A60, 47A57, 15A04, Secondary 12D15, 45P05, 47B38 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. |
| title | On Positivity Preservers with constant Coefficients and their Generators |
| topic | Algebraic Geometry Functional Analysis Primary 44A60, 47A57, 15A04, Secondary 12D15, 45P05, 47B38 |
| url | https://arxiv.org/abs/2308.10455 |