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The complexity on the family of operators. (English) Zbl 1363.47014
Summary: In this paper the complexity about the family of bounded linear operators is discussed. We first give a characterization about a family of operators on Banach space to have a dense $$G_\delta$$ set common hypercyclic vectors. Then as an application, we show that the set of all common hypercyclic vectors for the family $$\{\lambda B: \lambda \in \Lambda\}$$ is a dense $$G_\delta$$ set, where $$B$$ is the unilateral backward shift and $$\Lambda$$ is a bounded, closed subset of $$\mathbb{C}$$ with $$\lambda \in \Lambda$$, $$|\lambda| > 1$$.
##### MSC:
 47A16 Cyclic vectors, hypercyclic and chaotic operators