Several problems are considered in the setting of Hilbert spaces of holomorphic functions on the unit disc. In Chapter 1, the main result is a characterization of the zero sets of a large class of such spaces. The characterization is in terms of Gram matrices of reproducing kernels associated with the points of a sequence in the disc. The construction involved in the proof is then applied to a smaller class of spaces to characterize the elements of a space having a fixed inner function as a factor. Specializing to the case of the Dirichlet space, the construction gives a characterization of the boundary-zero sets.
This construction gives rise to a wandering vector of the shift operator of multiplication by z. In the case of the Dirichlet space and certain generalizations, the subspaces invariant under the shift operator are generated by wandering vectors. Furthermore, the wandering subspace of an invariant subspace is one-dimensional. Therefore, the problem of describing the invariant subspaces of Dirichlet-type spaces reduces to the problem of describing the wandering vectors. For invariant subspaces of the Dirichlet space determined by zero sets, inner divisors, and boundary-zero sets, the aforementioned construction produces a wandering vector and generator. It is not known whether other invariant subspaces of the Dirichlet space exist.
In Chapter 2, variants of the classical Toeplitz operators on H2 are studied. A characterization is obtained for the bounded, harmonic symbols giving rise to a bounded Toeplitz operator on a Dirichlet-type space. The relationship between the characterizing condition and multipliers of the holomorphic and harmonic Dirichlet spaces is examined.