Spectral properties of invariant subspaces in the Bergman space (Q1316407)

The author studies the relationship between two types of spectra associated with invariant subspaces of the Bergman space \(L^ 2_ a(\mathbb{D})\) and the function theoretic properties of the invariant subspaces themselves. For instance, he proves that if an invariant subspace \(J\) contains a function that is bounded away from 0 on some neighborhood of a point \(\lambda\) on the unit circle \(\mathbb{T}\), then the spectrum of \(z[J]\), multiplication by \(z\), when regarded as operating on the quotient space \(L^ 2_ a(\mathbb{D})/J\), does not contain the point \(\lambda\). 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.