Characterization of Matrix $K$-Positivity Preserver for $K=\mathbb{R}^n$ and for Compact Sets $K\subseteq\mathbb{R}^n$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914221965443072 |
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| author | di Dio, Philipp J. Langer, Lars-Luca |
| author_facet | di Dio, Philipp J. Langer, Lars-Luca |
| contents | For any closed $K\subseteq\mathbb{R}^n$, in [P.\ J.\ di\,Dio, K.\ Schmüdgen: $K$-Positivity Preserver and their Generators, SIAM J.\ Appl.\ Algebra Geom.\ 9 (2025), 794--824] all $K$-positivity preserver have been characterized, i.e., all linear maps $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ such that $Tp\geq 0$ on $K$ for all $p\geq 0$ on $K$. An important extension of polynomials $\mathbb{R}[x_1,\dots,x_n]$ with real coefficients are polynomials $\mathbb{R}^{m\times m}[x_1,\dots,x_n]$ with matrix coefficients. Non-negativity on $K$ for matrix polynomials with Hermitian coefficients $\mathrm{Herm}_m$ is then $p(x)\succeq 0$ for all $x\in K$. In the current work, we investigate linear maps $T:\mathrm{Herm}_m[x_1,\dots,x_n]\to\mathrm{Herm}_m[x_1,\dots,x_n]$. We focus on matrix $K$-positivity preserver, i.e., $Tp\succeq 0$ on $K$ for all $p\succeq 0$ on $K$. For $K=\mathbb{R}^n$ and compact sets $K\subseteq\mathrm{R}^n$, we give characterizations of matrix $K$-positivity preservers. We discuss the difference between the real and the matrix coefficient case and where our proof fails for general sets $K\subseteq\mathbb{R}^n$ with $K\neq \mathbb{R}^n$ and $K$ non-compact. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22584 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterization of Matrix $K$-Positivity Preserver for $K=\mathbb{R}^n$ and for Compact Sets $K\subseteq\mathbb{R}^n$ di Dio, Philipp J. Langer, Lars-Luca Functional Analysis Algebraic Geometry Primary 44A60, 47A57, Secondary 47B38, 60E07 For any closed $K\subseteq\mathbb{R}^n$, in [P.\ J.\ di\,Dio, K.\ Schmüdgen: $K$-Positivity Preserver and their Generators, SIAM J.\ Appl.\ Algebra Geom.\ 9 (2025), 794--824] all $K$-positivity preserver have been characterized, i.e., all linear maps $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ such that $Tp\geq 0$ on $K$ for all $p\geq 0$ on $K$. An important extension of polynomials $\mathbb{R}[x_1,\dots,x_n]$ with real coefficients are polynomials $\mathbb{R}^{m\times m}[x_1,\dots,x_n]$ with matrix coefficients. Non-negativity on $K$ for matrix polynomials with Hermitian coefficients $\mathrm{Herm}_m$ is then $p(x)\succeq 0$ for all $x\in K$. In the current work, we investigate linear maps $T:\mathrm{Herm}_m[x_1,\dots,x_n]\to\mathrm{Herm}_m[x_1,\dots,x_n]$. We focus on matrix $K$-positivity preserver, i.e., $Tp\succeq 0$ on $K$ for all $p\succeq 0$ on $K$. For $K=\mathbb{R}^n$ and compact sets $K\subseteq\mathrm{R}^n$, we give characterizations of matrix $K$-positivity preservers. We discuss the difference between the real and the matrix coefficient case and where our proof fails for general sets $K\subseteq\mathbb{R}^n$ with $K\neq \mathbb{R}^n$ and $K$ non-compact. |
| title | Characterization of Matrix $K$-Positivity Preserver for $K=\mathbb{R}^n$ and for Compact Sets $K\subseteq\mathbb{R}^n$ |
| topic | Functional Analysis Algebraic Geometry Primary 44A60, 47A57, Secondary 47B38, 60E07 |
| url | https://arxiv.org/abs/2512.22584 |