Hypercyclic subspaces for sequences of finite order differential operators

It is proved that, if (Pₙ) 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 c-dimensional subspace of entire functions, all of whose nonzero members are hypercyclic for the corresponding sequence (Pₙ(D)) of differential operators. In both cases, the subspace can be chosen so as to contain any prescribed hypercylic function.