Characterization of Matrix $K$-Positivity Preserver for $K=\mathbb{R}^n$ and for Compact Sets $K\subseteq\mathbb{R}^n$
Philipp J. di Dio, Lars-Luca Langer
TL;DR
The paper advances the theory of positivity preservers by extending the Borcea–Di Dio–Schmüdgen framework from scalar polynomials to Hermitian matrix polynomials. It develops a canonical operator representation $T=\sum_{\alpha\in\mathbb{N}_0^n} \frac{1}{\alpha!}\,(Q_\alpha\times \partial^\alpha)$ and proves equivalences linking $K$-positivity preservation for $K=\mathbb{R}^n$ and compact $K$ to convolution-type integral representations with matrix-valued measures. For $K=\mathbb{R}^n$, positivity preservation is tied to shifted operators $T_y$ and to measures $\mu_{y,M}$ governing $T_{y,M}$, with $Q_\beta(M)(y)$ given by moments of these measures; however, indeterminacy can prevent positivity of derived measures in the general (noncompact) setting. When $K$ is compact, the results strengthen: there exist positive operator-valued measures $\nu_y$ on $K-y$ representing the action via $T_y p(x)=\int_{K-y} p(x+t)\,d\nu_y(t)$ and $Q_\beta(M)(y)=\int t^\beta \,d\nu_y[M](t)$, ensuring positivity and determinacy. Overall, the work generalizes scalar results to matrix polynomials, clarifying the role of operator-valued measures and moment problems in positivity preservation.
Abstract
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.