Mathematical Sciences: Holomorphic Spaces

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McCarthy will study various spaces of holomorphic functions and the natural operators acting on these spaces. He will investigate the existence of radial or non-tangential boundary values for every function in an infinite dimensional Hilbert or Banach holomorphic space, and the connection between this phenomenon and the geometry of the space. He will also investigate how elements of holomorphic spaces can be decomposed into products of special functions, analogous to the inner-outer factoring in the classical Hardy spaces. Operator theory is that part of mathematics that studies the infinite dimensional generalizations of matrices. In particular, when restricted to finite dimensional subspaces, an operator has the usual linear properties, and thus can be represented by a matrix. The central problem in operator theory is to classify operators satisfying additional conditions given in terms of associated operators (e.g. the adjoint) or in terms of the underlying space. Operator theory underlies much of mathematics, and many of the applications of mathematics to other sciences.