截断Toeplitz 算子因为在算子理论、函数理论等领域的重要作用由Sarason首先系统研究并迅速成为近几年函数空间算子理论中研究的热点。本项目主要利用Banach代数局部化理论、模型空间的再生核和Berezin变换理论、截断Toeplitz 算子与截断Hankel算子和奇异积分算子等之间的联系研究如下问题：1.对于两个截断Toeplitz 算子研究算子符号满足什么分析、几何和代数性质可以保证它们的换位子和半换位子是紧或有限秩；2. 对有限个截断Toeplitz 算子研究它们的符号满足什么分析、几何和代数性质能够保证这些截断Toeplitz 算子的乘积还是截断Toeplitz 算子以及这些截断Toeplitz 算子的乘积是截断Toeplitz 算子的紧扰动或有限秩扰动。 3. 研究以逐段连续函数为符号的截断Toeplitz 算子Fredholm性质、指标及其生成的C*-代数的结构。

Truncated Toeplitz operators play an important role in operator theory and function theory,etc.Since Sarason has published his seminal paper,truncated Toeplitz operators have constituted an active area of research in operator theory.In this project,by using Banach algebra localization method,reproducing kernels and Berezin transformations,and the relationship between truncated Toeplitz operators and truncated Hankel operators and a class of singular integral operators,we will investigated:1.For two truncated Toeplitz operators, What analysis、algebraic and geometric properties of their symbols can guarantee that their commutators or semicommutators are compact; 2.For finite number truncated Toeplitz operators, What analysis、algebraic and geometric properties of their symbols can guarantee that a product of these truncated Toeplitz operators is itself a truncated Toeplitz operator or a product of these truncated Toeplitz operators is a compact(finite rank) perturbation of a truncated Toeplitz operator;3.Fredholm properties and indices of truncated Toeplitz operators with piecewise continuous function symbols and the structure of C*-algebra generated by truncated Toeplitz operators with piecewise continuous function symbols.