Quasi-Universal Functions for Sequences of Composition Operators on H 2 Via Hoffman’s Theory

Abstract

Let φ be a non-elliptic automorphism of the unit disk D. Gallardo and Gorkin (respectively Gallardo, Gorkin and Suárez) showed that there exists an interpolating Blaschke product b (respectively a thin Blaschke product) such that the linear span of its orbit under the composition operator C φ is dense in H 2 (in other words that b is a cyclic vector for C φ ). Using Hoffman's theory, we extend their result from the automorphic case to the selfmap case and show that for any sequence (φn) of holomorphic selfmaps of D with |φn(0)| → 1 and lim sup n→∞ |φ n (0)| 1 − |φn(0)| 2 = 1, there exists a thin Blaschke product B with zero distribution as sparse as one wishes such that the linear span of the orbit {B • φn : n ∈ N} of B is dense in H 2 . We will dub such a function a quasi-universal function. We also describe the limit points in the topological space M (H ∞ ) D of these admissible sequences (φn). This approach will enable us to give shorter proofs of the Gallardo-Gorkin result as well as the result of Bayart, Gorkin, Grivaux and the author on the existence of B-universal functions for admissible sequences. Finally we discuss the problem whether interpolating Blaschke products can ever be B-universal for admissible sequences (φn).Mathematics Subject Classification (2010). Primary 47B33; Secondary 46J15, 47A16, 30J10.