Mathematical Sciences: Structure of Toeplitz and CompositionOperators

考恩教授将研究某些有界的 单位圆盘的哈代空间上的线性算子， 具体介绍了运营商，以便深入了解更多 一般问题。他将继续调查 复合算子和Toeplitz算子，主要问题 就是识别交换子，决定次正规性和亚正规性 这里设想的研究是关于解析函数的， 数学的中心思想超过了一个世纪。 它们可以被不同地定义为某个简单的 偏微分方程系统，作为映射， 平面区域到其他平面区域， 角，除非在孤立的奇点，或作为限制 在适当精确的意义上的多项式。对应于 不同的描述方式，解析函数可以是 以各种方式进行研究。在这方面， 该项目是算子理论，建立希尔伯特空间， 函数的多项式和他们的极限在一个特定的 时尚，然后研究出现的空间上的运算符 自然地从函数中，例如通过乘法或通过 混合物.相反，这些函数理论运算符通常 在算子理论中是相当普遍的，所以通过研究 函数中可以学到很多关于运算符的知识， 抽象的。

Professor Cowen will study the structure of certain bounded linear operators on the Hardy space of the unit disk, looking at concretely presented operators in order to gain insight into more general problems. He will continue his investigations of composition operators and Toeplitz operators, the main issues being to identify commutants and decide sub- and hyponormality. 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.