Abstract
We study hypercyclicity properties of functions of Banach space operators. Generalizations of the results of Herzog–Schmoeger and Bermudez–Miller are obtained. As a corollary we also show that each non-trivial operator commuting with a generalized backward shift is supercyclic. This gives a positive answer to a conjecture of Godefroy and Shapiro. Furthermore, we show that the norm-closures of the set of all hypercyclic (mixing, chaotic, frequently hypercyclic, respectively) operators on a Hilbert space coincide. This implies that the set of all hypercyclic operators that do not satisfy the hypercyclicity criterion is rather small—of first category (in the norm-closure of hypercyclic operators).
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