Mathematical Sciences: Matrix Analysis and Toeplitz and Composition Operators

考恩将研究单位圆盘上解析函数的Hardy Hilbert空间、Bergman Hilbert空间和相关空间上的有界线性算子的结构，探讨具体提出的算子上的问题，以便深入了解更一般的问题。他将致力于将Bergman和Hardy空间上关于复合算子的结果统一到解析函数空间上的复合算子理论中，强调空间之间的相似性，并最大限度地减少对特定空间特殊特征的依赖。算子论是研究矩阵的无限维广义的数学部分。特别地，当被限制在有限维子空间时，算子具有通常的线性性质，因此可以用矩阵来表示。算子理论中的中心问题是根据伴随算子(例如伴随算子)或基础空间对满足附加条件的算子进行分类。算符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。

Cowen will study the structure of bounded linear operators on the Hardy Hilbert space, the Bergman Hilbert space, and related spaces of analytic functions on the unit disk, approaching problems on concretely presented operators in order to gain insight into more general problems. He will work on unifying results concerning composition operators on Bergman and Hardy spaces into a theory of composition operators on spaces of analytic functions emphasizing similarities between spaces and minimizing reliance on special features of particular 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.