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Operators with dense, invariant, cyclic vector manifolds. (English) Zbl 0732.47016
Let X be a Banach space, T a bounded operator on X, and \(x\in X\). The vector x is hypercyclic for T if the set \({\mathcal O}_ x=\{x,Tx,T^ 2x,...\}\) is dense in X; x is supercyclic if \(\cup_{\lambda \in {\mathbb{C}}}\lambda {\mathcal O}_ x\) is dense in X; x is cyclic if the linear space generated by \({\mathcal O}_ x\) is dense in X. The phenomenon of hypercyclicity was regarded as a somewhat unusual thing, and few examples of hypercyclic vectors were known (see the references in this article). The authors give a general study of the phenomenon, and construct remarkable classes of operators which have numerous hypercyclic vectors. For instance, they show that there are certain operators S such that every operator \(T\neq 0\) commuting with S has a dense linear manifold consisting entirely of hypercyclic vectors (except the origin). The methods used are quite elementary and very ingenious.
The authors point out that their results are not relevant for the invariant subspace problem since all of the operators they construct have many invariant subspaces. They point out to some connections between the material of the paper and the theory of chaotic transformations.

MSC:
47A65 Structure theory of linear operators
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