TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 1, Pages 247–267 S 0002-9947(00)02743-4 Article electronically published on September 13, 2000 SPECTRAL THEORY AND HYPERCYCLIC SUBSPACES FERNANDO LE ´ ON-SAAVEDRA AND ALFONSO MONTES-RODR ´ IGUEZ Abstract. A vector x in a Hilbert space H is called hypercyclic for a bounded operator T : H→H if the orbit {T n x : n ≥ 1} is dense in 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) → 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. 1. Introduction A bounded operator T on a Hilbert space H is said to be cyclic if there is a vector x ∈H such that the orbit {T n x} n≥1 has dense linear span. If this is the case, the vector x is called a cyclic vector for T . If the orbit {T n x} n≥1 is itself dense in H, then T is said to be hypercyclic. In this case the vector x is called hypercyclic for T . Each of the following classes of linear maps contains hypercyclic operators: back- ward and bilateral shifts [Ro], [GS], [Sa2], translations and differentiation operators [CS], composition operators [BS1], [BS2], multiplication operators [GoS], perturba- tion of the identity by a weighted shift [Sa2]. Interest in cyclic operators arises from the invariant subspace problem. In fact, it is easy to see that an operator T has no non-trivial invariant closed subspace if and only if each non-zero vector is cyclic for T . It is not known if there is a bounded linear operator on a separable Hilbert space that does not have closed, non-trivial invariant subspaces. Similarly, an operator has no non-trivial closed invariant subset if and only if each non-zero vector is hypercyclic. Again, it is not known if there is an operator on Hilbert that does not have closed, non-trivial Received by the editors April 14, 1997. 2000 Mathematics Subject Classification. Primary 47A16, 47A53; Secondary 47B37. Key words and phrases. Hypercyclic operator, hypercyclic vector, essential spectrum, essential minimum modulus, bilateral shift, backward shift, multiplier, composition operator, differentiation operator, translation operator. c 2000 American Mathematical Society 247 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use