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Hypercyclicity for translations through Runge’s theorem. (English) Zbl 1142.47010
A bounded linear operator \(T\) acting on a Fréchet space \(X\) is said to be hypercyclic with a hypercyclic vector \(x\in X\) if the orbit \(\{T^nx:n\geq0\}\) is dense in \(X\). Baire’s theorem ensures that any countable family of hypercyclic operators has a common hypercyclic vector. Yet, of course, an uncountable family of hypercyclic operators may or may not have a common hypercyclic vector; see, for instance, [F. Bayart, J. Oper. Theory 52, No. 2, 353–370 (2004; Zbl 1104.47009)].
For every \(\alpha\in\mathbb{C}\), let \(T_\alpha\) be the translation operator defined on the space \(H(\mathbb{C})\) of all entire functions by \(T_\alpha(f):z\mapsto f(\alpha+z)\). In [Adv. Math. 182, No. 2, 278–306 (2004; Zbl 1066.47005)], G. Costakis and M. Sambarino used the classical Runge theorem to show that the uncountable family \((T_\alpha)_{\alpha\not=0}\) has a common hypercyclic vector. In the paper under review, the author establishes a vector-valued version of Runge’s theorem and applies it to extend Costakis and Sambarino’s result to uncountable families of translations on separable subspaces \(X\) of \(H_b (E)\) satisfying certain conditions. Here, \(H_b(E)\) is the Fréchet space of all entire functions of bounded type on a Banach space \(E\).

MSC:
47A16 Cyclic vectors, hypercyclic and chaotic operators
46G20 Infinite-dimensional holomorphy
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