Supercyclic subspaces: Spectral theory and weighted shifts

Abstract

A vector x in a Banach space B is called hypercyclic for a bounded operator T if the orbit {Tnx: n ≥ 0} is dense in B. If the scalar multiples of the elements in the orbit are dense, then the vector x is called supercyclic. We give a general sufficient condition for a bounded operator on a Banach space to have an infinite-dimensional closed subspace of supercyclic vectors. As a consequence, we also obtain a spectral sufficient condition for the existence of such a subspace for an operator. These results allow us to characterize unilateral and bilateral weighted shifts that have an infinite dimensional closed subspace of supercyclic vectors. Surprisingly, there are weighted shift operators that have supercyclic vectors, but in which all closed subspaces of supercyclic vectors are finite dimensional. Our results complement recent work on hypercyclic subspaces and supercyclic subspaces. They also suggest that the problem of the existence of infinite dimensional closed subspaces of supercyclic vectors is not only different, but also more difficult to handle than the corresponding problem for hypercyclic vectors. © 2001 Academic Press.