Zero sets of interpolating Blaschke products

Abstract

For a function h in //, Z(h) denotes the zero set of h in the maximal ideal space of H + C. It is well known that if q is an interpolating Blaschke product then Z(q) is an interpolation set for H. The purpose of this paper is to study the converse of the above result. Our theorem is: If a function h is in // and Z(h) is an interpolation set for 7/, then there is an interpolating Blaschke product q such that Z(q) = Z(). As applications, we will study that for a given interpolating Blaschke product q, which closed subsets of Z(q) are zero sets for some functions in H. We will also give a characterization of a pair of interpolating Blaschke products q and q 2 such that Z(q x ) U Z(q 2 ) is an interpolation set for i/ 00 .