Rings of PDE-preserving operators on nuclearly entire functions

• Autorzy: Henrik Petersson
• Języki publikacji: EN

Abstrakty

EN

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 𝒪(ℙ) for ℙ forms a ring and, as a starting point, we characterize 𝒪(ℍ) in different ways, where ℍ = ℍ(F) is the set of continuous homogeneous polynomials on F. The elements of 𝒪(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 𝒪(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).

Kategorie tematyczne

• 46G20: Infinite-dimensional holomorphy
• 47L10: Algebras of operators on Banach spaces and other topological linear spaces
• 47B99: None of the above, but in this section
• 47F05: Partial differential operators (should also be assigned at least one other classification number in section 47)
• 47A15: Invariant subspaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2004
• Tom: 163
• Numer: 3
• Strony: 217-229

Daty

wydano

2004

Twórcy

autor

Henrik Petersson

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

Identyfikatory

DOI

10.4064/sm163-3-2