本项目致力于函数空间上的Toeplitz算子及其相关算子的代数性质的研究，属于多复变函数论及算子理论中的前沿热点课题。我们将研究Bergman空间、调和Bergman空间以及Segal-Bergman空间上的Toeplitz算子及其相关算子的交换性、准交换性、零乘积问题、有限秩问题、以及两个Toeplitz算子乘积什么条件下能够等于另外一个Toeplitz算子等基本性质，并进而研究Toeplitz代数及其相关问题。同以往在记号函数调和或多重调和的前提下进行研究的思路不同，本项目将首先探讨k-拟齐次Toeplitz算子的基本性质，然后利用极分解式最终得到一般Toeplitz算子的代数性质，进一步揭示单变量与多变量、不同函数空间上算子理论的联系与不同。

This project is dedicated to the research of algebraic properties of Toeplitz operators and related operators, which is a hot topics in multicomplex function theory and operator theory. We will study some basic algebraic properties of Toeplitz operators and related operators on the Bergman space, harmonic Bergman space, Segal-Bergman space, including the commutativity, semi-commutativity, zero-product problem, finite-rank problem and when the product of two Toeplitz operators is a Toeplitz operator, and then we will explore the Toeplitz algebra and related issues. Different from the previous research ideas which is on the premise of that the symbol is harmonic or pluriharmonic, this project will first explore some basic properties of k-quasihomogeneous Toeplitz operators. Then we use a polar decomposition to get the algebraic properties of general Toeplitz operators, in order to further reveal the relation and difference between single variables and several variables, and operator theory in different function spaces.