Rings of PDE-preserving operators on nuclearly entire functions
Volume 163 / 2004
Abstract
Let $E,F$ be Banach spaces where $F=E'$ or vice versa. If $F$ has the approximation property, then the space of nuclearly entire functions of bounded type, ${\scr H}_{\rm Nb}(E)$, and the space of exponential type functions, ${\rm Exp} (F)$, form a dual pair. The set of convolution operators on ${\scr H}_{\rm Nb} (E)$ (i.e. the continuous operators that commute with all translations) is formed by the transposes $\varphi (D) \equiv{} ^{t}{\varphi}$, $\varphi\in {\rm Exp} (F)$, of the multiplication operators $\varphi :\psi \mapsto \varphi \psi $ on ${\rm Exp} (F)$. A continuous operator $T$ on ${\scr H}_{\rm Nb} (E)$ is PDE-preserving for a set ${\mathbb P} \subseteq {\rm Exp} (F)$ if it has the invariance property: $T\ker \varphi (D)\subseteq \ker \varphi(D)$, $\varphi\in {\mathbb P}$. The set of PDE-preserving operators ${\scr O} ({\mathbb P})$ for ${\mathbb P}$ forms a ring and, as a starting point, we characterize ${\scr O}({\mathbb H})$ in different ways, where ${\mathbb H}={\mathbb H} (F)$ is the set of continuous homogeneous polynomials on $F$. The elements of ${\scr O} ({\mathbb H})$ can, in a one-to-one way, be identified with sequences of certain growth in ${\rm Exp} (F)$. Further, we establish a kernel theorem: For every continuous linear operator on ${\scr H}_{\rm Nb} (E)$ there is a unique kernel, or symbol, and we characterize ${\scr O} ({\mathbb H})$ by describing the corresponding symbol set. We obtain a sufficient condition for an operator to be PDE-preserving for a set ${\mathbb P}\supseteq {\mathbb H} $. Finally, by duality we obtain results on operators that preserve ideals in ${\rm Exp}(F)$.
