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Hypercyclic and cyclic vectors. (English) Zbl 0853.47013
Let $$\mathcal X$$ denote a separable complex Banach space. A vector $$x\in {\mathcal X}$$ is said to be hypercyclic for an operator $$T$$ on $$\mathcal X$$ if the set $$\{T^n x: n\in \mathbb{N}\}$$ is norm dense in $$\mathcal X$$. We say that $$x$$ is supercyclic if the set $$\{aT^n x: n\in \mathbb{N}, a\in \mathbb{C}\}$$ is norm dense. An operator is called hypercyclic (supercyclic) if it has a hypercyclic (supercyclic) vector.
B. Beauzamy [On the orbits of a linear operator, preprint] and C. Kitai [Invariant closed sets for linear operators, Thesis, University of Toronto (1982)] have shown that the set of all hypercyclic (supercyclic) vectors for a hypercyclic (supercyclic) operator on a Banach space is norm dense. B. Sz-Nagy and C. Foiaş [Stud. Math. 31, 35-42 (1968; Zbl 0174.18203)] have shown that the set of cyclic vectors is norm dense when the underlying space is Hilbert.
In the present article it is shown that if $$T$$ is a cyclic operator on $$\mathcal X$$ such that the point spectrum of $$T^*$$ contains no nonempty open set then the set of all cyclic vectors is dense. A result of K. F. Clancey and D. D. Rogers [Indiana Univ. Math. J. 27, 689-696 (1978; Zbl 0396.47016)] on the density of cyclic vectors is also generalized. It is also shown that if a vector $$x$$ in $$\mathcal X$$ is hypercyclic for $$T$$ then $$x$$ is hypercyclic for all $$T^n$$ for all $$n\geq 1$$.

##### MSC:
 47A65 Structure theory of linear operators
##### Keywords:
hypercyclic; supercyclic; density of cyclic vectors
##### Citations:
Zbl 0174.18203; Zbl 0396.47016
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