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On weakly transitive operators. (English) Zbl 1244.47009

A bounded linear operator \(T:X\to X\) acting on a Banach space \(X\) is called weakly hypercyclic if there exists a vector \(x\in X\) such that its orbit under \(T\), i.e., the set \(\{ T^nx\,:\, n\geq 0\}\) is weakly dense in \(X\) and \(T:X\to X\) acting on a Banach space \(X\) with separable dual is called weakly transitive on a nonempty subset \(D\) of \(X\) if, for every nonempty norm open set \(U\subset X\) and every nonempty weakly open set \(V\subset X\) such that \(U\cap D\neq\emptyset\) and \(V\cap D\neq\emptyset\), there exists a non-negative integer \(n\) such that \(T^n(U\cap D)\cap V\cap D\neq\emptyset\). The author shows that, if an operator \(T\) is weakly transitive on a bounded norm-open subset \(D\subset X\) such that the set \(\cup_{n\geq 0}T^nD\) is weakly dense in \(X\), then \(T\) is weakly hypercyclic on \(X\). Hence, if \(T\) is weakly transitive on the open unit ball of \(X\), then it is weakly hypercyclic on \(X\). Furthermore, the author gives an example of a bilateral weighted shift on \(l^2(\mathbb{Z})\) which is weakly hypercyclic but fails to be weakly transitive on any bounded open subset of \(l^2(\mathbb{Z})\).

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46B45 Banach sequence spaces
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