Maximal Cluster Sets of L-Analytic Functions Along Arbitrary Curves

Abstract

Let Ω be a domain in the N -dimensional real space, L be an elliptic differential operator, and (T n ) be a sequence whose members belong to a certain class of operators defined on the space of L-analytic functions on Ω. It is proved in this paper the existence of a dense linear manifold of L-analytic functions all of whose nonzero members have maximal cluster sets under the action of every T n along any curve ending at the boundary of Ω such that its ω-limit does not contain any component of the boundary. The above class contains all partial differentiation operators ∂ α , hence the statement extends earlier results due to Boivin, Gauthier and Paramonov, and to the first, third and fourth authors. Key words and phrases: maximal cluster set, L-analytic function, dense linear manifold, admissible path, elliptic operator, internally controlled operator.

Interpolation by hypercyclic functions for differential operators

“…Lemma 2.3 says that the differential operators induced by entire functions of subexponential type are "internally controlled". Its easy proof is sketched in [4]. Finally, Lemma 2.4 furnishes a statement about the strong linear structure of a topologically mixing sequence of linear mappings.…”

Maximal cluster sets on spaces of holomorphic functions

“…We point out that Theorem 2.9 has been recently extended (see [6]) to domains in R N having no connected component consisting of a single point, to elliptic differential operators L and to certain class of operators T acting on the space of L-analytic functions; the holomorphic case and the operators T = D n are contained as special cases.…”

“…To end this section, we remark that it is possible to extend the searching of large sets of functions with maximal cluster sets along arbitrary curves to more general settings (see for instance [14] and [6]). …”

Universal Taylor series with maximal cluster sets

“…Next, we present a concept that was introduced by the authors in [12]. If G ⊂ C is a domain, then we say that an operator (= linear continuous self-map) T :…”

“…(In [36] the set A was a segment, but the proof would be the same for our more general family; note also that the class Γ 0 (G) contains strictly the class Γ(G) of curves γ ⊂ G tending to the boundary with non-total oscillation, that is, such that (∂G) \ γ = ∅; the class Γ(G) is considered in [12] and [14], and a glance at the proofs reveals that the results in these two papers hold for Γ 0 (G) instead of Γ(G).) (See also [15,Theorem 2.1] for an extension of the Kierst-Szpilrajn statement where certain holomorphic operators participate.)…”