Invariant Subspaces in Bergman and Dirichlet Spaces

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Abstract 9706905 Richter Richter and Sundberg will investigate the structure of the invariant subspace lattices of bounded analytic functions acting as multiplication operators on Banach or Hilbert spaces of analytic functions. The knowledge about the invariant subspaces of the unilateral shift (i.e. multiplication by z on the Hardy space of the unit disc) has been used via dilation and model theory to enhance our knowledge about contraction operators on Hilbert spaces. The spaces that will be considered include the Dirichlet and Bergman spaces of the unit disc. The operator theoretic questions that arise are naturally related to questions from classical analysis, such as questions about zero sets and sets of uniqueness. The study of linear operators acting on Hilbert spaces can be viewed as an extension of linear algebra to an infinite dimensional setting. It has historical origins and applications in such fields as partial differential equations, fluid dynamics, and quantum mechanics. The work of Richter and Sundberg is concerned with certain natural models for such operators. It is expected that a study of these models will yield insight into general structural properties of Hilbert space operators.