×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite