Beurling's Theorem for the Bergman space

It is interesting to study Hilbert spaces of analytic functions on the unit disk D and the operators on them. One of the most elementary of these operators is multiplication by the coordinate function z. The Hardy space H² constructs an important example in this area. A famous theorem by A. Beurling characterizes the invariant subspaces of H², and thus the invariant subspaces of the unilateral shift. In this thesis, we will be interested in the Bergman space L²a. It has been known for some time that the invariant subspace lattice of L2 a is very complicated when compared to that of H². In particular, not every invariant subspace of L²a is cyclic. Nevertheless, the following analogue of Beurling's Theorem is true and is the main result of this thesis: Theorem. Let M be an invariant subspace of L²a. Then: M = [M Ɵ zM]. Thus, as in the Hardy space case, invariant subspaces of L²a are in a one-to-one correspondence with their wandering subspaces.