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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
##### Keywords:
hypercyclic operators; translations; Runge’s theorem
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