Hypercyclic weighted shifts

Abstract

An operator T acting on a Hilbert space is hypercyclic if, for some vector x in the space, the orbit {T"x: n > 0} is dense.In this paper we characterize hypercyclic weighted shifts in terms of their weight sequences and identify the direct sums of hypercyclic weighted shifts which are also hypercyclic.As a consequence, we show within the class of weighted shifts that multi-hypercyclic shifts and direct sums of fixed hypercyclic shifts are both hypercyclic.For general hypercyclic operators the corresponding questions were posed by D. A. Herrero, and they still remain open.Using a different technique we prove that / + T is hypercyclic whenever T is a unilateral backward weighted shift, thus answering in more generality a question recently posed by