Rings of PDE-preserving operators on nuclearly entire functions

Abstract

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 → ϕψ 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: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 ).

Polynomial projectors preserving homogeneous partial differential equations

“…Suppose that (c s ) |s|=k , c s ∈ C, is a sequence such that |s|=k c s w s = 0. This means that F is a polynomial of degree d − k that is identically equal to zero due to (9). Hence, we proved that if for (c s ) |s|=k , c s ∈ C, there holds (8), then we have…”

“…In [8] (see also [9]) Petersson has settled a convenient formalism (using the concept of pairing of Banach spaces) and extended results of [3] to Banach spaces. Our Theorem 1 is likely to have a similar infinite dimensional counterpart.…”

Hypercyclic sequences of PDE-preserving operators

“…Then Im K ⊆ P m so Proposition 13 applies and we note that I − K = n>m H n . (Note that K ∈ O(H), and the projectors in O(H) onto the spaces P m , for which Proposition 13 thus applies, are described in [26]. )…”

“…PDE-preserving operators have been studied in [26][27][28]. In particular, by H we denote the set of homogeneous polynomials, and we have the following characterization result for O(H).…”

Entire Functions on Infinite Dimensional Spaces

“…Dwyer in [2] considers also partial differential operators of infinite order on spaces of formal series which correspond via the Taylor series transformation to spaces of entire functions of exponential type, see [10]. Recently Petersson in [12,13] has considered operators on spaces of entire functions, which preserve partial differential equations in a certain sense.…”