Journal of the
Korean Mathematical Society JKMS

A continuous linear operator $T$, on the space of entire functions in $d$ variables, is PDE-preserving for a given set $\mathbb P \subseteq \mathbb C [\xi_1 ,...,\xi_d]$ of polynomials if it maps every kernel-set $\ker P(D)$, $P\in \mathbb P$, invariantly. It is clear that the set ${\mathscr O} (\mathbb P)$ of PDE-preserving operators for $\mathbb P$ forms an algebra under composition. We study and link properties and structures on the operator side $\mathscr O (\mathbb P)$ versus the corresponding family $\mathbb P$ of polynomials. For our purposes, we introduce notions such as the PDE-$preserving$ $hull$ and $basic$ $sets$ for a given set $\mathbb P$ which, roughly, is the largest, respectively a minimal, collection of polynomials that generate all the PDE-preserving operators for $\mathbb P$. We also describe PDE-preserving operators via a $kernel$ theorem. We apply Hilbert's Nullstellensatz.

Keywords: PDE-preserving, PDE-preserving hull, basic, convolution operator, exponential type, Fourier-Borel transform, algebra, invariant, Hilbert's Nullstellensatz.

MSC numbers: Primary 47A15, 47L10, 47L15; Secondary 08A99, 08A40, 47F05