Hypercyclic weighted shifts. (English) Zbl 0822.47030

Summary: An operator \(T\) acting on a Hilbert space is hypercyclic if, for some vector \(x\) in the space, the orbit \(\{T^ n x: n\geq 0\}\) 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 multihypercyclic 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.


47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47B99 Special classes of linear operators
Full Text: DOI


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