Hypercyclic subspaces for sequences of finite order differential operators

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.