Mathematical Sciences: Subnormal Operators and Banach Spaces of Holomorphic Functions

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Abstract of the project (item C) The principal investigator will continue to study the internal structure theory of subnormal operators, with particular consideration of finite rank self-commutator, the structure of invariant subspaces, and connections and applications to other fields. He will use operator theoretical tools to study the theory of quadrature domains. He also plans to investigate the structure of invariant subspaces of the Bergman space on the unit disk and multipliers between the Bergman spaces over a general bounded region. Operator theory is a central discipline in modern mathematics that studies infinite dimensional generalization of matrices. This type of research is an attempt to classify operators satisfying additional conditions given in terms of the associated operator or the underlying space. They have rich applications in every applied science as well as pure mathematics. To mention a specific example, one of basic H-infinity control problems was developed by the classical mathematical theory of Nevanlinna-Pick interpolation. Applications of the theory include improvement in automobile suspension systems, airplane flight control, chemical process control, etc.