On the hereditarily hypercyclic operators. (English) Zbl 1114.47008

A linear and continuous operator \(T:X\to X\) on a separable Banach space X is said to be hypercyclic whenever there exists a vector \(x\in X\) with dense orbit \(\{T^nx:\;n\geq0\}\) in \(X\). Given an increasing sequence \((n_k)\) of positive integers, \(T\) is said to be hereditarily hypercyclic (HHC) with respect to \((n_k)\) if \((T^{m_k})\) is hypercyclic for every subsequence \((m_k)\) of \((n_k)\).
The paper under review deals mainly with this special case of hypercyclicity. In particular, it is proved that a linear continuous operator \(T\) is HHC with respect to \((n_k)\) if and only if given two non-void open subsets \(U, V\) of \(X\), \(T^{n_k}(U)\cap V\neq\varnothing\) for any \(k\) large enough; and if \(T\) is HHC with respect to a syndetic sequence \((n_k)\) (that is, \(\sup_k(n_{k+1}-n_k)<\infty\)), then it is HHC with respect to the whole sequence. In addition, applications to the bilateral backward shift operator on the space \(L^p(\beta)\) are given.


47A16 Cyclic vectors, hypercyclic and chaotic operators
47L10 Algebras of operators on Banach spaces and other topological linear spaces
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