Interpolating Blaschke Products and Factorization Theorems

Abstract

Let M(H∞) be the maximal ideal space of H∞ the Banach algebra of bounded analytic functions on the open unit disk. Let G be the set of nontrivial points in M(H∞). By Hoffman's work, G has deep connections with the zero sets of interpolating Blaschke products. It is proved that for a closed ρ-separated subset E of M(H∞) with E ⊂ G, there exists an interpolating Blaschke product whose zero set contains E. This is a generalization of Lingenberg's theorem. Let f be a continuous function on M(H∞). Suppose that f is analytic on a nontrivial Gleason part P(x), f(x) = 0, and f ≠ 0 on P(x). It is proved that there is an interpolating Blaschke product b with zeros {zn}n such that b(x) = 0 and f(zn) = 0 for every n. This fact can be used for factorization theorems in Douglas algebras and in algebras of functions analytic on Gleason parts.