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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
Citations:
Zbl 1082.47004
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