Classification of reducing subspaces of a class of multiplication operators on the Bergman space via the Hardy space of the bidisk

ABSTRACT. In this paper we obtain a complete description of nontrivial minimal reduc-ing subspaces of the multiplication operator by a Blaschke product with four zeros on the Bergman space of the unit disk via the Hardy space of the bidisk. Let D be the open unit disk in C. Let dA denote Lebesgue area measure on the unit disk D, normalized so that the measure of D equals 1. The Bergman space L2a is the Hilbert space consisting of the analytic functions on D that are also in the space L2(D, dA) of square integrable functions on D. For a bounded analytic function φ on the unit disk, the multiplication operator Mφ with symbol φ is defined on the Bergman space L2a given by Mφh = φh for h ∈ L2a. On the basis {en}∞n=0, where en is equal to n+ 1zn, the multiplication operator Mz by z is a weighted shift operator, said to be the Bergman shift