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

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)].