Mathematical Sciences: Invariant Subspaces in some Banach of Analytic Functions

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9401027 Sundberg Richter and Sundberg will investigate the invariant sbspace lattices of the operator of multiplication by an analytic function on Banach or Hilbert spaces of analytic functions. The spaces that will be investigated include the Dirichlet and Bergman spaces on the open unit disc, as well as Hardy spaces of more general domains in the complex plane. These invariant subspace problems are related naturally to questions from classical analysis, such as questions about zero sets and sets of uniqueness. The new technique involved in these investigation will be that of "extremal functions" in invariant subspaces. Operator theory is that branch of mathematics that treats objects that are infinite generalizations of finite matrices. Frequently, these operators can be given realization as mutiplications on certain collections of functions. In this form the invariant subspace and classification problem are related to natural function existence and interpolation questions. This project will contribute to both the classification problem and the related concrete function existence problems. ***