This paper is concerned with the behavior of a singular inner function on the maximal ideal space of the algebra of bounded analytic functions in the unit disk, specifically with their zero sets confined to the maximal ideal space minus the unit disk. While the motivation for this specific problem comes from a question posed by Guillori and Sarason, the paper is actually part of the author’s general program of studying the behavior of bounded analytic functions on the maximal ideal space. In particular, it is a continuation of his recent work [“Common zero sets of equivalent singular inner functions”, Stud. Math. 163, No. 3, 231–255 (2004; Zbl 1074.46034)]; both parts contain plenty of useful material.
The paper begins with a new factorization result for inner functions (Theorem 2.1) which states that an inner function with a disconnected zero set (split into two components) can be written as the product of two inner functions having precisely these components as their zero sets. This theorem is used to deduce some interesting properties of the connected components of the common zero set of equivalent singular inner functions. Among other results, the author shows that for an inner function with at least two singularities on the unit circle, its zero set minus the disk is disconnected.