Rings of PDE-preserving operators on nuclearly entire functions

Henrik Petersson¹

¹ School of Mathematical Sciences Chalmers/Göteborg University Eklandagatan, SE-412 96 Göteborg, Sweden

Studia Mathematica, Tome 163 (2004) no. 3, pp. 217-229

Résumé

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 ${\rm Exp}(F)$.

Détail

DOI : 10.4064/sm163-3-2

Keywords: banach spaces where vice versa has approximation property space nuclearly entire functions bounded type scr space exponential type functions exp form dual pair set convolution operators scr continuous operators commute translations formed transposes varphi equiv varphi varphi exp multiplication operators varphi psi mapsto varphi psi exp continuous operator scr pde preserving set mathbb subseteq exp has invariance property ker varphi subseteq ker varphi varphi mathbb set pde preserving operators scr mathbb mathbb forms ring starting point characterize scr mathbb different ways where mathbb mathbb set continuous homogeneous polynomials elements scr mathbb one to one identified sequences certain growth exp further establish kernel theorem every continuous linear operator scr there unique kernel symbol characterize scr mathbb describing corresponding symbol set obtain sufficient condition operator pde preserving set mathbb supseteq mathbb finally duality obtain results operators preserve ideals exp