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Some necessary and sufficient conditions for hypercyclicity criterion. (English) Zbl 1084.47003

The authors give necessary and sufficient conditions for a linear continuous operator \(T\) on an infinite-dimensional separable Hilbert space \(H\) to satisfy the hypercyclicity criterion, which states that \(T: H\to H\) is hypercyclic if there exist dense subsets \(Y\), \(Z\) of \(H\), a sequence \((n_k)\), and functions \(S_{n_k}: Z\to X\) such that \(T^{n_k}y\to 0\) \((y\in Y)\), \(S_{n_k}z\to 0\) \((z\in Z)\), and \(T^{n_k}S_{n_k}z\to z\) \((z\in Z)\). Stimulated by the open question whether each hypercyclic operator in \(B(H)\) satisfies this criterion, the authors prove that \(T\in B(H)\) satisfies the hypercyclicity criterion if and only if for each pair \(U\), \(V\) of nonempty open subsets of \(H\) and each open neighborhood \(W\) of zero, \(T^n U\cap W\neq\emptyset\) and \(T^n W\cap V\neq\emptyset\) for some integer \(n\). Based on this result, further characterizations are given.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
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