HYPERCYCLIC WEIGHTED SHIFTS

An operator T acting on a Hilbert space is hypercyclic if, for some vector x in the space, the orbit {T(n)x: n greater than or equal to 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 I + T is hypercyclic whenever T is a unilateral backward weighted shift, thus answering in more generality a question recently posed by K. C. Chan and J. H. Shapiro.