Mathematical Sciences: Toeplitz Operators and Hilbert Spaces of Analytic Functions

麦卡锡教授的研究项目将涵盖希尔伯特空间算子理论和复数分析的许多领域：(A)他将Toeplitz算子理论的某些方面推广到对数模代数的设置。(B)他将研究Toeplitz算子在解析函数的Hilbert空间上的有界性。(C)他将研究涉及多个复变量中的Toeplitz算子的问题。希尔伯特空间算子本质上是复数的无限矩阵。这些运算符在应用科学的各个领域都有应用，在纯数学中也是如此。这种类型的研究是对此类物体的某些重要家族进行分类的一种尝试。

Professor McCarthy's research project will cover a number of areas in Hilbert space operator theory and complex analysis: (a) He will extend some aspects of the theory of Toeplitz operators to the setting of logmodular algebras. (b) He will investigate the boundedness of Toeplitz operators on Hilbert spaces of analytic functions. (c) He will study questions dealing with Toeplitz operators in several complex variables. Hilbert space operators are essentially infinite matrices of complex numbers. These operators have applications in every area of applied science as well as in pure mathematics. This type of research is an attempt to classify certain important families of such objects.