Infinite dimensional universal subspaces generated by Blaschke products

Abstract:

Let $H^\infty$ be the Banach algebra of all bounded analytic functions in the unit disk $\mathbb D$. A function $f\in H^\infty$ is said to be universal with respect to the sequence $(\frac {z+z_n}{1+\overline {z}_nz})_n$ of noneuclidian translates, if the set $\{f(\frac {z+z_n}{1+\overline {z}_nz}):n\in \mathbb {N}\}$ is locally uniformly dense in the set of all holomorphic functions bounded by $||f||_\infty$. We show that for any sequence of points $(z_n)$ in $\mathbb {D}$ tending to the boundary there exists a closed subspace of $H^\infty$, topologically generated by Blaschke products, and linear isometric to $\ell ^1$, such that all of its elements $f$ are universal with respect to noneuclidian translates. The proof is based on certain interpolation problems in the corona of $H^\infty$. Results on cyclicity of composition operators in $H^2$ are deduced.