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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.

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