Existence and non-existence of frequently hypercyclic subspaces for weighted shifts

TL;DR

This paper investigates the conditions under which weighted shift operators on Banach spaces have frequently hypercyclic subspaces, providing new examples and answering an open problem in the field.

Contribution

It presents the first examples of weighted shifts with and without frequently hypercyclic subspaces, resolving an open question from prior research.

Findings

Constructed a weighted shift on lp with a frequently hypercyclic subspace

Provided an example of a weighted shift with a hypercyclic but not frequently hypercyclic subspace

Answered positively an open problem regarding the existence of such subspaces

Abstract

We study the existence and the non-existence of frequently hypercyclic subspaces in Banach spaces. In particular, we give an example of a weighted shift on lp possessing a frequently hypercyclic subspace and an example of a frequently hypercyclic weighted shift on lp possessing a hypercyclic subspace but no frequently hypercyclic subspace. The latter example allows us to answer positively Problem 1 posed by Bonilla and Grosse-Erdmann in [Monatsh. Math. 168 (2012)].