Hypercyclic subspaces for Frechet space operators

A continuous linear operator T : X -> X is hypercyclic if there is an x is an element of X such that the orbit {T(n)x} is dense, and such a vector x is said to be hypercyclic for T. Recent progress show that it is possible to characterize Banach space operators that have a hypercyclic subspace, i.e., an infinite dimensional closed subspace H subset of X of, except for zero, hypercyclic vectors. The following is known to hold: A Banach space operator T has a hypercyclic subspace if there is a sequence (n(i)) and an infinite dimensional closed subspace E subset of X such that T is hereditarily hypercyclic for (n(i)) and T-ni -> 0 pointwise on E. In this note we extend this result to the setting of Frechet spaces that admit a continuous norm, and study some applications for important function spaces. As an application we also prove that any infinite dimensional separable Frechet space with a continuous norm admits an operator with a hypercyclic subspace. (c) 2005 Elsevier Inc. All rights reserved.