Mathematical Sciences: Operators on Spaces of Analytic Functions

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DMS-9622914 PI: William Ross University of Richmond The principal investigator will conduct research on operators on spaces of analytic functions. Such problems explore the relationship between functional analysis and complex function theory. The first part of the proposal examines the invariant subspaces of the "backward shift operator" on a general Hilbert space of analytic functions. In the second part of the proposal, the principal investigator plans to examine the Bergman space on a "crescent" domain and examine certain subspaces which are invariant under the linear transformation "multiplication by z". In the third part of the proposal, the principal investigator plans to look at the closure of the polynomials on the "standard crescent" (the region bounded by two internally tangent circles). As mentioned earlier, these types of problems combine two fields of mathematics, functional analysis and complex function theory. This link was first noticed by Arne Beurling in 1949 in his seminar paper on the shift operator. Further results on the shift were taken up by several Russians such as Korenblum, Nikolskii, Havin, Shirokov, and Shamoyan. In this proposal we examine these same problems but will pay closer attention to the geometry involved.