On Positivity Preserving, Translation Invariant, Operators in $L^p(\mathbb{R}^n)^m$

TL;DR

This paper characterizes a class of linear operators in vector-valued Lebesgue spaces that preserve positivity and are invariant under translation, providing a comprehensive mathematical description of their structure.

Contribution

It offers a complete characterization of positivity preserving, translation invariant linear operators in $L^p(R^n)^m$, advancing the understanding of their properties in functional analysis.

Findings

Provides a mathematical description of such operators.

Identifies conditions under which operators preserve positivity.

Establishes the structure of translation invariant operators.

Abstract

We characterize positivity preserving, translation invariant, linear operators in $L^p(\mathbb{R}^n)^m$, $p \in [1,\infty)$, $m,n \in \mathbb{N}$.