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Linear Operators $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ and $K$-Positivity Preserver: A Short Review

Philipp J. di Dio

TL;DR

This paper surveys the emerging theory of linear operators on the polynomial ring $\mathbb{R}[x_1,\dots,x_n]$ that preserve nonnegativity on a closed set $K$. It adopts a functional-analytic framework based on moment functionals, Fréchet and LF-spaces, and regular Fréchet Lie groups to derive canonical representations of operators and criteria for generating positivity-preserving dynamics. Key contributions include a canonical differential-operator form for $T$, a characterization of $K$-positivity preservers via moment sequences, and a complete description of generators for $K$-positivity preserving semigroups in both constant- and non-constant coefficient settings (the latter via pointwise Lévy-type representations). The results connect real algebraic geometry with infinite-dimensional analysis, providing a foundational framework for further study of linear operators on polynomial rings under positivity constraints and inviting extensions to broader classes of sets $K$ and operators.

Abstract

In the current short review we present the latest developments on linear maps $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$, especially of $K$-positivity preserver, i.e., $Tp\geq 0$ on $K\subseteq\mathbb{R}^n$ for all $p\in\mathbb{R}[x_1,\dots,x_n]$ with $p\geq 0$ on $K$.

Linear Operators $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ and $K$-Positivity Preserver: A Short Review

TL;DR

This paper surveys the emerging theory of linear operators on the polynomial ring that preserve nonnegativity on a closed set . It adopts a functional-analytic framework based on moment functionals, Fréchet and LF-spaces, and regular Fréchet Lie groups to derive canonical representations of operators and criteria for generating positivity-preserving dynamics. Key contributions include a canonical differential-operator form for , a characterization of -positivity preservers via moment sequences, and a complete description of generators for -positivity preserving semigroups in both constant- and non-constant coefficient settings (the latter via pointwise Lévy-type representations). The results connect real algebraic geometry with infinite-dimensional analysis, providing a foundational framework for further study of linear operators on polynomial rings under positivity constraints and inviting extensions to broader classes of sets and operators.

Abstract

In the current short review we present the latest developments on linear maps , especially of -positivity preserver, i.e., on for all with on .
Paper Structure (10 sections, 13 theorems, 56 equations)

This paper contains 10 sections, 13 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Moment Functionals and Sequences
  4. Frechét and LF-Spaces
  5. Regular Frechét Lie Groups
  6. Representations of Linear Operators on $\mathds{R}[x_1,\dots,x_n]$
  7. $K$-Positivity Preservers
  8. Generators $A$ of Linear Maps $T$: $T=e^{A}$
  9. Generators $A$ of $K$-Positivity Preserving Semi-Groups $(e^{tA})_{t\geq 0}$
  10. Summary

Key Result

Theorem 3.1

Let $n\in\mathds{N}$ and let be linear. Then, for all $\alpha\in\mathds{N}_0^n$, there exist unique polynomials $q_\alpha\in\mathds{R}[x_1,\dots,x_n]$ such that

Theorems & Definitions (24)

  • Example 2.1: see e.g. treves67
  • Definition 2.2: see e.g. omori97
  • Theorem 3.1: folklore, canonical representation
  • Lemma 3.2: didio25KPosPresGen
  • Lemma 3.3: didio25hadamardLanger
  • Corollary 3.4: didio25hadamardLanger
  • Definition 4.1
  • Example 4.2
  • Example 4.3: didio25hadamardLanger
  • Definition 4.4
  • ...and 14 more