Rings of PDE-preserving operators on nuclearly entire functions

Henrik Petersson

Studia Mathematica, Tome 162 (2004), p. 217-229 / Harvested from The Polish Digital Mathematics Library

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, $ℋ_{Nb}(E)$, and the space of exponential type functions, Exp(F), form a dual pair. The set of convolution operators on $ℋ_{Nb}(E)$ (i.e. the continuous operators that commute with all translations) is formed by the transposes $φ(D) ≡ ^{t}φ$, φ ∈ Exp(F), of the multiplication operators φ :ψ ↦ φ ψ on Exp(F). A continuous operator T on $ℋ_{Nb}(E)$ is PDE-preserving for a set ℙ ⊆ Exp(F) if it has the invariance property: T ker φ(D) ⊆ ker φ(D), φ ∈ ℙ. The set of PDE-preserving operators (ℙ) for ℙ forms a ring and, as a starting point, we characterize (ℍ) in different ways, where ℍ = ℍ(F) is the set of continuous homogeneous polynomials on F. The elements of (ℍ) can, in a one-to-one way, be identified with sequences of certain growth in Exp(F). Further, we establish a kernel theorem: For every continuous linear operator on $ℋ_{Nb}(E)$ there is a unique kernel, or symbol, and we characterize (ℍ) by describing the corresponding symbol set. We obtain a sufficient condition for an operator to be PDE-preserving for a set ℙ ⊇ ℍ. Finally, by duality we obtain results on operators that preserve ideals in Exp(F).

Détail

Publié le : 2004-01-01

Zbl 1065.46029

EUDML-ID : urn:eudml:doc:285046