多变量函数空间上算子理论与单变量函数空间上算子理论有本质的差别。由于多变量函数空间的复杂性，使其上算子理论的研究更具有吸引力。推广的多变量Bergman 空间是关于更一般测度下的Bergman 空间，通常的多变量Bergman空间是它的一种特殊情形。研究经验表明通常Bergman 上算子理论的问题，可能在推广Bergman 空间上，利用更高级的方法得到解决，推广的多变量Bergman 空间上的算子理论是一个全新的研究领域。本项目主要研究如下三个问题：（1） 推广的多变量Bergman空间上Toeplitz 算子的交换C*-代数的刻画，给出生成交换C*-代数的Toeplitz 算子符号特征。（2）推广的多变量Bergman空间上Toeplitz 算子和Hankel的亚正规性、有限秩、零积等问题的研究。（3）推广的Bergman空间上的算子在工程技术科学、特别是在控制理论中的应用

Function theory of several variables is essentially different from that of single variable. Operator theory on several variables function space has more attractiveness because of complexity of several variables function space. Generalized several variables Bergman space is more general Bergman space with respect to a general measure, and usual Bergman spaces is special Generalized several variables Bergman space. Recent some researches have shown that some problems on usual Bergman spaces may be sloved on Generalized Bergman space by more higher methods. Operator theory on Generalized several variables Bergman space is a new area of study, and it is remarkable that many simple questions remain still unsolved. This research consists of three parts: (1) Characterize commutative C*-algebra generated by Toeplitz operators on Generalized several variables Bergman space,and give symbol features of Toeplitz operator which generate commutative C*-algebra. (2)study hypormality、 zero products problem and finite rank problem of Toeplitz operators on Generalized several variables Bergman space.(3)study application of Toeplitz oprators in control theory.