Mathematical Sciences: Analytic Function Spaces

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9401234 Yang This award supports mathematical research investigating spaces of analytic functions and the operator obtained through multiplication by the identity function. The goals are to study the existence of nontangential limits for each function in the closure of the analytic polynomials in some power norm relativeto a measure and the properties of the shift operator on these spaces. In addition work will be done investigating the structure of invariant subspaces and self-duality. Further work focuses on the set of bounded point evaluations for the closure of rational functions with poles off a given compact set, approximation problems for "near" analytic function spaces and the spectral properties of the shift acting on the closure of theideal generated by the conjugate of the identity. The study of function spaces obtained as the closure of aclass of polynomials has a long history going back before Mergalyan. Its value derives from a combination of approximation results stated in more or less geometric terms. The domain of these investigations is confined to the use of analytic functions, requiring considerable expertise in the field of classical function theory and the modern field of Banach spaces. ***