Rings of PDE-preserving operators on nuclearly entire functions
Let E;F be Banach spaces where F = E 0 or vice versa. If F has the approximation property, then the space of nuclearly entire functions of bounded type, HNb(E), and the space of exponential type functions, Exp(F ), form a dual pair. The set of convolution operators on HNb(E) (i.e. the continuous operators that commute with all translations) is formed by the transposes '(D) t ', '2 Exp(F ), of the multiplication operators ' : 7! ' on Exp(F ). A continuous operator T on HNb(E) is PDE-preserving for a set P Exp(F ) if it has the invariance property: T ker'(D) ker'(D), ' 2 P. The set of PDE-preserving operators O(P) for P forms a ring and, as a starting point, we characterize O(H) in dieren t ways, where H = H(F ) is the set of continuous homo- geneous polynomials on F. The elements of O(H) can, in a one-to-one way, be identied with sequences of certain growth in Exp(F ). Further, we establish a kernel theorem: For every continuous linear operator on HNb(E) there is a unique kernel, or symbol, and we characterize O(H) by describing the corresponding symbol set. We obtain a sucien t con- dition for an operator to be PDE-preserving for a setP H. Finally, by duality we obtain results on operators that preserve ideals in Exp(F ).