Hypercyclic subspaces for Fréchet space operators ✩

A continuous linear operator T :X →X is hypercyclic if there is an x ∈X such that the orbit {T nx} 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 ⊆ 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 (ni ) and an infinite dimensional closed subspace E ⊆X such that T is hereditarily hypercyclic for (ni ) and T ni → 0 pointwise on E. In this note we extend this result to the setting of Fréchet 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 Fréchet space with a continuous norm admits an operator with a hypercyclic subspace. © 2005 Elsevier Inc. All rights reserved.