Limits of hypercyclic and supercyclic operators. (English) Zbl 0758.47016

Let \(T\) be a bounded linear operator on a complex, separable infinite dimensional Hilbert space \(H\), and let \(y\in H\). The orbit of \(y\) under \(T\) is the set \(\text{Orb}(T,y):=\{y,Ty,T^ 2y,\dots\}\). A vector \(y\in H\) is called hypercyclic (resp., supercyclic) for \(T\), if \(\text{Orb}(T,y)\) is dense in \(H\) (resp., if the set of scalar multiples of \(\text{Orb}(T,y)\) is dense in \(H\)). The author obtains a spectral characterization of the norm closure of the class of all hypercyclic operators on \(H\), and describes the structure of the set of all hypercyclic vectors of a given hypercyclic operator. Analogous results are obtained for supercyclic operators and vectors.


47A65 Structure theory of linear operators
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