Let T be a bounded linear operator acting on a complex, separable, infinite-dimensional Hilbert space and let f: D → ℂ be an analytic function defined on an open set D ⊆ ℂ which contains the spectrum of T. If T is the limit of hypercyclic operators and if f is nonconstant on every connected component of D, then f(T) is the limit of hypercyclic operators if and only if $f(σ_{W}(T)) ∪ {z ∈ ℂ: |z| = 1}$ is connected, where $σ_{W}(T)$ denotes the Weyl spectrum of T.
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Extending previous results of H. Salas we obtain a characterisation of hypercyclic weighted shifts on an arbitrary F-sequence space in which the canonical unit vectors $(e_n)$ form a Schauder basis. If the basis is unconditional we give a characterisation of those hypercyclic weighted shifts that are even chaotic.
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