PDE-PRESERVING PROPERTIES

Abstract

A continuous linear operator T, on the space of entire functions in d variables, is PDE-preserving for a given set <TEX>$\mathbb{P}\;\subseteq\;\mathbb{C}|\xi_{1},\ldots,\xi_{d}|$</TEX> of polynomials if it maps every kernel-set ker P(D), <TEX>$P\;{\in}\;\mathbb{P}$</TEX>, invariantly. It is clear that the set <TEX>$\mathbb{O}({\mathbb{P}})$</TEX> of PDE-preserving operators for <TEX>$\mathbb{P}$</TEX> forms an algebra under composition. We study and link properties and structures on the operator side <TEX>$\mathbb{O}({\mathbb{P}})$</TEX> versus the corresponding family <TEX>$\mathbb{P}$</TEX> of polynomials. For our purposes, we introduce notions such as the PDE-preserving hull and basic sets for a given set <TEX>$\mathbb{P}$</TEX> which, roughly, is the largest, respectively a minimal, collection of polynomials that generate all the PDE-preserving operators for <TEX>$\mathbb{P}$</TEX>. We also describe PDE-preserving operators via a kernel theorem. We apply Hilbert's Nullstellensatz.