Mathematical Sciences: Holomorphic Spaces

McCarthy将研究全纯函数的各种空间和作用于这些空间的自然算子。他将研究无限维Hilbert或Banach全纯空间中每个函数的径向或非切向边值的存在性，以及这种现象与空间几何之间的联系。他还将研究如何将全纯空间的元素分解成特殊函数的乘积，类似于经典Hardy空间中的内外因式分解。算符理论是研究矩阵的无限维推广的数学部分。特别地，当限制在有限维子空间中时，算子具有通常的线性性质，因此可以用矩阵表示。算子理论的中心问题是对满足附加条件的算子进行分类，这些附加条件是根据相关算子（如伴随算子）或根据基础空间给出的。算子理论是许多数学的基础，也是数学在其他科学中的许多应用的基础。

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.