Non-Finite Dimensional Closed Vector Spaces of Universal Functions for Composition Operators

Abstract

AbstractLet H(Ω) be the space of analytic functions on a complex region Ω, which is not the punctured plane. In this paper, we prove that if a sequence of automorphisms {φn}n ≥ 0 of Ω has the property that for every compact subset K ⊂ Ω there is a positive integer n such that K ∩ φn(K) = 0, then there exists an infinite dimensional closed vector subspace F ⊂ H(Ω) such that for all f ∈ F\{0} the orbit (f ∘ φn)n ≥ 0 is dense in H(Ω). The corresponding result for the punctured plane is somewhat different and is also studied