Mathematical Sciences: Problems in Function-Theoretic Operator Theory

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Professor Kriete will investigate four problems in which operator theory and function theory interact in a fruitful way. The first of these is to determine completely the invariant subspace lattice of a particular cyclic subnormal operator related to the Cesaro operator. The second concerns composition operators acting on Bergman spaces. The third is a classical question in weighted polynomial approximation which is of interest in the study of subnormal operators. The last is about the extremal behavior of the norms in the de Branges - Rovnyak functional model for contraction operators. The research envisioned here is about analytic functions, a central preoccupation of mathematics for well over a century. They can be defined variously as solutions of a certain simple system of partial differential equations, as maps which take planar regions to other planar regions in a way that preserves angles except at isolated singularities, or as limits of polynomials in a suitably precise sense. Corresponding to the diverse ways of describing them, analytic functions can be studied in a variety of ways. The approach favored in this project is operator-theoretic, building up Hilbert spaces of functions by taking polynomials and their limits in a particular fashion, then studying operators on the spaces that arise naturally from the functions, e.g. by multiplication or by composition. Conversely, these function-theoretic operators often turn out to be quite general in operator theory, so by studying the functions one can learn a great deal about operators in the abstract. Professor Kriete's project is to travel this two-way street in both directions.