Non-finite dimensional closed vector spaces of universal functions for composition operators
For a domain \(\Omega\) in \(\mathbb{C}\), \(H (\Omega)\) denotes, as usual, the space of holomorphic functions on \(\Omega\), endowed with the topology of uniform convergence on compact subsets, and \(\Aut (\Omega)\) stands for the set of automorphisms of \(\Omega\). A sequence \((\varphi_n)_{n \geq 0}\) in \(\Aut (\Omega)\) is said to be a run-away sequence if for each compact subset \(K\) of \(\Omega\) there exists a positive integer \(n_0 = n_0 (K)\) such that \(K \cap \varphi_{n_0} (K) = \emptyset\). The authors prove that if \(\Omega\) is a domain in \(\mathbb{C}\) which is not the punctured plane, i.e. not isomorphic to \(\mathbb{C} \backslash \{0\}\), and if \((\varphi_n)_{n \geq 0}\) is a run-away sequence in \(\Aut (\Omega)\), then there exists an infinite dimensional closed vector subspace \(F\) of \(H (\Omega)\) such that the orbit \((f \circ \varphi_x)_{n \geq 0}\) is dense in \(H (\Omega)\) for all \(f \in F \backslash \{0\}\). The corresponding result when \(\Omega\) is the punctured plane is somewhat different and is also studied by the authors.
scientific article; zbMATH DE number 803702