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

Abstract

Let \({\phi}\) be a non-elliptic automorphism of the unit disk \({\mathbb D}\). Gallardo and Gorkin (respectively Gallardo, Gorkin and Suarez) showed that there exists an interpolating Blaschke products b (respectively a thin Blaschke product) such that the linear span of its orbit under the composition operator \({C_\phi}\) is dense in H2 (in other words that b is a cyclic vector for \({C_\phi}\)). Using Hoffman’s theory, we extend their result from the automorphic case to the selfmap case and show that for any sequence \({(\phi_n)}\) of holomorphic selfmaps of \({\mathbb D}\) with \({|\phi_n(0)|\to 1}\) and $$\begin{array}{ll}\limsup\limits_{n\to\infty}\frac{|\phi_n'(0)|}{1-|\phi_n(0)|^2}=1,\end{array}$$ there exists a thin Blaschke product B with zero distribution as sparse as one wishes such that the linear span of the orbit \({\{B\circ \phi_n:n\in\mathbb N\}}\) of B is dense in H2. We will dub such a function a quasi-universal function. We also describe the limit points in the topological space \({M(H^\infty)^\mathbb D}\) of these admissible sequences \({(\phi_n)}\). 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 \({\fancyscript B}\) -universal functions for admissible sequences. Finally we discuss the problem whether interpolating Blaschke products can ever be \({\fancyscript B}\) -universal for admissible sequences \({(\phi_n)}\).