Multiplication invariant subspaces of the Bergman space

A theorem of Aleman, Richter, and Sundberg asserts that every z-invariant subspace M of the Bergman space A(2) is generated by M circle minus zM, the orthocomplement of zM within M. The purpose of this paper is investigate the extent to which that property generalizes to g-invariant subspaces for a function g E H. Such a function g is said to have the wandering property in A(2) if every g-invariant subspace M of A(2) is generated by M circle minus gM. In the Hardy space H-2, every inner function has the wandering property, while every function with this property must be the composition of an inner function with a conformal mapping. In this paper it is shown that the only functions that can have the wandering property in A(2) are essentially the classical inner functions. On the other hand, a large class of inner functions for which this property fails is exhibited. The wandering property is equivalent to the cyclicity of certain reproducing kernels; thus the proofs involve the approximation of noncyclic kernels by kernels corresponding to pullback measures.