Hypercyclic subspaces for sequences of finite order differential operators

TL;DR

This paper demonstrates the existence of large subspaces of entire functions that are hypercyclic for sequences of differential operators derived from polynomials with unbounded valences, extending hypercyclicity theory.

Contribution

It establishes the existence of infinite-dimensional and dense subspaces of entire functions that are hypercyclic for sequences of differential operators from polynomials with unbounded valences.

Findings

Existence of infinite-dimensional hypercyclic subspaces.

Existence of dense hypercyclic subspaces of continuum dimension.

Subspaces can contain any prescribed hypercyclic function.

Abstract

It is proved that, if $(P_n)$ is a sequence of polynomials with complex coefficients having unbounded valences and tending to infinity at sufficiently many points, then there is an infinite dimensional closed subspace of entire functions, as well a dense $\mathfrak{c}$-dimensional subspace of entire functions, all of whose nonzero members are hypercyclic for the corresponding sequence $(P_n(D))$ of differential operators. In both cases, the subspace can be chosen so as to contain any prescribed hypercylic function.