For a bounded region G ⊂ ℂ and a compact set K ⊂G , with area measure zero, we will characterize the invariant subspaces M (under ƒ → z ƒ) of the Bergman space Lpa(G\K), 1 ≤ p < ∞, which contain L<sup>pa(G) and with dim(M/(z-⋋)M) = 1 for all ⋋ ∈ G\K. When G\K is connected, we will see that dim(M/(z-⋋)M) = 1 for all ⋋ ∈ G\K and this in this case we will have a complete description of the invariant subspaces lying between L<sup>pa(G) and Lpa(G\K). When G\K is not connected, we will show that in general the invariant subspaces between Lpa(G) and Lpa(G\K) are fantastically complicated. As an application of these results, we will remark on the complexity on the invariant subspaces (under ƒ → ζ ƒ) of certain Besov spaces on K. In particular, we shall see that in the harmonic Dirichlet space B12(∏), there are invariant subspaces F such that the dimension of ζF in F is infinite.
Alexandru Aleman, Stefan Richter, and William T. Ross. Bergman Spaces on Disconnected Domains. Technical paper (TR-94-04). Math and Computer Science Technical Report Series. Richmond, Virginia: Department of Mathematics and Computer Science, University of Richmond, July 6, 1994.