Richter and Sundberg will continue their study of algebras of bounded analytic functions acting as multiplication operators on Banach or Hilbert spaces of analytic functions. For this project their main goal will be to extend current knowledge of invariant subspaces, cyclic vectors, and zero sets of Bergman, Dirichlet and related spaces in one variable, as well as of the Drury-Arveson-Hardy space on the unit ball of several complex variables.
The study of spaces of analytic functions has a long and rich history as a meeting ground and source of ideas from a wide range of areas in both Pure and Applied Mathematics such as Complex Analysis, Harmonic Analysis, Operator Theory, Functional Analysis, Control theory, and Partial Differential Equations. Operator theory has its roots in the work on Partial Differential Equations of Fredholm and Hilbert in the late nineteenth and early twentieth centuries. The study of spaces of analytic functions has its origins in the work on the classical Hardy spaces by Hardy, Fischer, and the Riesz brothers, among others, in the first half of the twentieth century. The two areas met in the 1940's in the work of A. Beurling on the unilateral shift, which yielded a complete structure theory of an important infinite dimensional operator using Hardy-space techniques. The generalization of Beurling's results to arbitrary multiplicity shifts together with the Sz.Nagy dilation theorem is the basis for a model theory for contraction operators on Hilbert spaces. Thus, up to scaling, every bounded linear operator on a separable Hilbert space can be modeled using a semi-invariant subspace of a vector-valued Hardy space. As many naturally occurring processes can be modeled by use of such linear operators, this has applications that can be felt throughout science. More research is needed and is being done to clarify the model theory. In particular, there has been a large effort devoted to extending the ideas developed in the study of the Hardy spaces to other spaces of analytic functions. The current work of Richter and Sundberg is in this area.
