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Rotations of hypercyclic and supercyclic operators. (English) Zbl 1079.47013
A (bounded linear) operator $$T$$ on a Banach space $$X$$ is called hypercyclic if there is a vector $$x \in X$$ such that its orbit $$\{T^n(x) \;| \;n=0,1,2,... \}$$ is dense in $$X$$; the vector $$x$$ is called hypercyclic for $$T$$. The operator $$T$$ is called supercyclic if $$\{ \alpha T^n(x) \;| \alpha \in \mathbb C, n \in \mathbb N \}$$ is dense in $$X$$; the vector $$x$$ is called supercyclic for $$T$$. Clearly, every hypercyclic operator is supercyclic. The authors prove the following nice result: If $$T$$ is a hypercyclic operator on $$X$$ and $$\lambda$$ is a complex number of modulus 1, then the operator $$\lambda T$$ is hypercyclic and has the same set of hypercyclic vectors as $$T$$. This is a consequence of a more general result which is obtained with a clever argument. As another consequence of the general result, they prove that if the point spectrum of the transpose $$T'$$ of the operator $$T$$ is empty, then the vector $$x$$ is supercyclic for $$T$$ if and only if the set $$\{t T^n(x) \;| \;t>0,\;n \in \mathbb N \}$$ is dense in $$X$$. These results answer several questions posed in the literature. A simple construction of a hypercyclic operator $$T$$ on a Hilbert space $$H$$ such that $$\lambda T$$ is not hypercyclic for each complex number of modulus different from one is exhibited. The main result of the paper is related to the so-called bounded steps problem for hypercyclic operators. Recently, A. Peris and L. Saldivia [Integral Equations Oper. Theory 51, 275–281 (2005; Zbl 1082.47004)], and S. Grivaux in her thesis independently, proved that an operator $$T$$ satisfies the hypercyclicity criterion if and only if for each strictly increasing sequence $$(n_k)_k$$ of natural numbers such that $$(n_{k+1}-n_k)_k$$ is bounded, there is a vector $$x \in X$$ such that $$\{T^{n_k}(x) \;| \;k \in \mathbb N \}$$ is dense in $$X$$.

MSC:
 47A16 Cyclic vectors, hypercyclic and chaotic operators
Zbl 1082.47004
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