Bulletin of the
Korean Mathematical Society BKMS

In this paper, we first adapt Runge's Theorem to work on certain domains in any complex Banach space. Then, using this result, we extend Birkhoff's Theorem on the hypercyclicity of translations on $H(\mathbb C)$ and Costakis' and Sambarino's result on the existence of common hypercyclic functions for uncountable families of translations on $H(\mathbb C)$ to subspaces of $H_b(E)$ (in some cases all of $H_b(E)$), $E$ being in a large class of Banach spaces.

Keywords: hypercyclic operators, hypercyclicity

MSC numbers: Primary 47A16, Secondary 46G20