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

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