Hypercyclic subspaces for sequences of finite order differential operators

TL;DR

The paper investigates when the set of entire functions that are hypercyclic for a sequence of differential operators $P_n(D)$, with $P_n$ polynomials of unbounded valence, contains large linear structures. By combining a general spaceability criterion for hypercyclic sequences, density results for exponential-type functions via $\mathcal{E}$-unicity sets, and constructive subspace techniques, the authors prove that $HC((P_n(D)))$ is spaceable, maximal dense-lineable, and, under a pointwise framework, pointwise $\mathfrak{c}$-spaceable and $\mathfrak{c}$-infinitely pointwise $\mathfrak{c}$-dense-lineable in $H(\mathbb{C})$. They show that one can embed any prescribed hypercyclic function into such subspaces, highlighting the robustness of hypercyclicity structures for operator sequences. The results extend known single-operator hypercyclicity lineability phenomena to sequences of finite-order differential operators, clarifying the role of unbounded valence and growth properties, and they pose questions about analogous phenomena for exponential-type operator families $\Phi_n(D)$.

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.

Theorems & Definitions (14)

• Theorem 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1
• proof
• Theorem 4.2