Hypercyclic subspaces and weighted shifts

We first generalize the results of León-Saavedra and Müller (2006) [10] on hypercyclic subspaces to sequences of operators on Fréchet 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 lplp or c0c0, on the space of entire functions and on certain Köthe 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)P(D) possesses a hypercyclic subspace.