Rings of PDE-preserving operators on nuclearly entire functions

Henrik Petersson. "Rings of PDE-preserving operators on nuclearly entire functions." Studia Mathematica 163.3 (2004): 217-229. <http://eudml.org/doc/285046>.

@article{HenrikPetersson2004,
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, $ℋ_\{Nb\}(E)$, 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).},
author = {Henrik Petersson},
journal = {Studia Mathematica},
keywords = {Banach spaces; dual pair; approximation property; invariance property; kernel theorem},
language = {eng},
number = {3},
pages = {217-229},
title = {Rings of PDE-preserving operators on nuclearly entire functions},
url = {http://eudml.org/doc/285046},
volume = {163},
year = {2004},
}

TY - JOUR
AU - Henrik Petersson
TI - Rings of PDE-preserving operators on nuclearly entire functions
JO - Studia Mathematica
PY - 2004
VL - 163
IS - 3
SP - 217
EP - 229
AB - 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).
LA - eng
KW - Banach spaces; dual pair; approximation property; invariance property; kernel theorem
UR - http://eudml.org/doc/285046
ER -