本项目主要研究函数空间上的算子理论中的有限秩问题，如多圆盘与球上的Hardy空间上的Toeplitz算子的交换子是有限秩算子的条件是什么？什么情况下，单圆盘Hardy空间上的几个Toeplitz算子的乘积模掉一个有限秩算子与乘积的顺序无关？单圆盘Hardy空间上的Toeplitz算子或Hankel算子的两两乘积的有限和为有限秩算子的充分必要条件是什么？多圆盘Bergman空间上具有有界n-调和函数符号的两个Toeplitz算子的乘积等于一个Toeplitz算子的有限秩扰动的充分必要条件是什么？多重调和Bergman空间与多重调和Hardy空间上的Toeplitz算子的半交换子与交换子为有限秩算子的条件是什么？等等。有限秩算子在算子理论与算子代数中有非常重要的作用，在Toeplitz算子理论中已经产生许多有重要影响的有限秩问题，我们的研究将进一步充实算子理论。

In this project, we main want to study these problems of finite rank in operator theory over function spaces. For example, what are the sufficient and necessary conditions for the commutator of Toeplitz operator be finite rank on the Hardy space of the ploydisk and unit ball ? Under what circumstances, the product of finite Toeplitz operators modulo a finite rank operator is unrelated with the order of the product in the unit disk; What are the sufficient and necessary conditions for the finite sum of pairwise product of Toplitz operator ang Hankel operator be finite rank operator in the unit disk? We will study the sufficient and necessary conditions for the product two Toeplitz operators with n-harmonic function symbols be equal to the finite rank perturbation of a Toeplitz operator. For semi-commutator and commutator of Toeplitz operators on pluriharmonic Bergman space and pluriharmonic Hardy space, what is the condition of those to be finite rank operator, and etc..The finite rank operator is very important in operator theory and operator algebra. There are many finite rank problems which have important influence in Toeplitz operator theory. Our research will be further enriching the theory of operator.