Infinite Dimensional Universal Subspaces Generated by Blaschke Products

LetH∞ be the Banach algebra of all bounded analytic functions in the unit disk D. A function f ∈ H∞ is said to be universal with respect to the sequence ( z+zn 1+znz )n of noneuclidian translates, if the set {f( z+zn 1+znz ) : n ∈ N} is locally uniformly dense in the set of all holomorphic functions bounded by ||f ||∞. We show that for any sequence of points (zn) in D tending to the boundary there exists a closed subspace of H∞, topologically generated by Blaschke products, and linear isometric to 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∞. Results on cyclicity of composition operators in H2 are deduced. During the last two decades problems on cyclicity of operators and universality of functions have gained more and more interest. The achievements are best described in the survey papers [5] and [6] of Grosse-Erdmann. In this paper we study universal Blaschke products, continuing research begun by Heins [7] and complementing known results [2], [1] and [4] on the existence of cyclic vectors for the parabolic and hyperbolic composition operators on the Hardy space H. A first extension of Heins’ result was given in [4], where it was shown that for every sequence (zn) tending to the boundary of the unit disk D = {z ∈ C : |z| < 1} there exists a Blaschke product B universal for noneuclidian translates, meaning that the set {B ( (z+zn)/(1+znz) ) : n ∈ N} is locally uniformly dense in the setB of all analytic functions in D bounded by one. This gave rise to a Blaschke product that was a joint cyclic vector for a given sequence of composition operators Cφn on H. Recall that a vector x is called cyclic for an operator T defined on a Fréchet space X, if the linear span of the orbit {Tx : x ∈ X} is dense in X. Here the symbols φn were chosen from the set of hyperbolic and parabolic automorphisms of the unit disk. In another direction, Aron and Gorkin [1] showed that in the Banach algebra H(Bn) of the ball Bn in C (n ≥ 1), there exists a closed, infinite-dimensional subspace such that all its norm one elements are universal with respect to some sequence of automorphisms. For the reader’s convenience, we shall define an extended concept of universal elements. Received by the editors September 6, 2005 and, in revised form, February 5, 2006. 2000 Mathematics Subject Classification. Primary 30D50; Secondary 47B33, 46J15, 30H05.