Spectral theory and hypercyclic subspaces

Abstract:

A vector $x$ in a Hilbert space $\mathcal {H}$ is called hypercyclic for a bounded operator $T: \mathcal {H} \rightarrow \mathcal {H}$ if the orbit $\{T^{n} x : n \geq 1 \}$ is dense in $\mathcal {H}$. Our main result states that if $T$ satisfies the Hypercyclicity Criterion and the essential spectrum intersects the closed unit disk, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors for $T$. The converse is true even if $T$ is a hypercyclic operator which does not satisfy the Hypercyclicity Criterion. As a consequence, other characterizations are obtained for an operator $T$ to have an infinite-dimensional closed subspace of hypercyclic vectors. These results apply to most of the hypercyclic operators that have appeared in the literature. In particular, they apply to bilateral and backward weighted shifts, perturbations of the identity by backward weighted shifts, multiplication operators and composition operators. The main result also applies to the differentiation operator and the translation operator $T:f(z)\rightarrow f(z+1)$ defined on certain Hilbert spaces consisting of entire functions. We also obtain a spectral characterization of the norm-closure of the class of hypercyclic operators which have an infinite-dimensional closed subspace of hypercyclic vectors.