Hypercyclic subspaces and weighted shifts

We first generalize the results of Le\'on and M\"uller [Studia Math. 175(1) 2006] on hypercyclic subspaces to sequences of operators on Fr\'echet spaces with a continuous norm. Then we study the particular case of iterates of an operator T and show a simple criterion for having no hypercyclic subspace. Finally we deduce from this criterion a characterization of weighted shifts with hypercyclic subspaces on the spaces lp or c0, on the space of entire functions and on certain K\"othe sequence spaces. We also prove that if P is a non-constant polynomial and D is the differentiation operator on the space of entire functions then P(D) possesses a hypercyclic subspace.