Hypercyclic and cyclic vectors.

*(English)*Zbl 0853.47013Let \(\mathcal X\) 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 \(n\geq 1\).

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 \(n\geq 1\).

Reviewer: K.Seddighi (Shiraz)

##### MSC:

47A65 | Structure theory of linear operators |