Mathematical Sciences: Structure of Toeplitz and CompositionOperators

考恩教授将研究单位圆盘的Hardy空间上某些有界线性算子的结构，考察具体的算子，以深入了解更一般的问题。他将继续他对复合算子和Toeplitz算子的研究，主要问题是识别交换子并判定次正规和次正规。这里设想的研究是关于解析函数的，这是一个多世纪以来数学的核心问题。它们可以被不同地定义为某一简单偏微分方程组的解，或者定义为将平面区域映射到其他平面区域的映射，该映射除了在孤立奇点处保持角度外，还可以定义为适当精确意义上的多项式的极限。与描述它们的不同方式相对应，解析函数可以用不同的方式来研究。这个项目中所采用的方法是算子论，通过以特定的方式获取多项式及其极限来建立函数的希尔伯特空间，然后研究由函数自然产生的空间上的算子，例如通过乘法或通过合成。相反，这些函数论算子在算子理论中往往是非常普遍的，所以通过研究这些函数，人们可以从抽象的角度了解大量关于算子的知识。

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.