Spectral Properties of Invariant Subspaces in the Bergman Space

We study the relationship between two types of spectra associated with invariant subspaces of the Bergman space L2a(D) and the function theoretic properties of the invariant subspaces themselves. For instance, we prove that if an invariant subspace J contains a function that is bounded away from 0 on some neighborhood of a point λ on the unit circle T, then the spectrum of z[J], multiplication by z when regarded as operating on the quotient space L2a(D)J, does not contain the point λ. A consequence of this result is that the spectrum associated with the invariant subspace of all functions vanishing on a prescribed Bergman space zero sequence coincides with the closure of the sequence.