函数空间上的算子理论因为与函数论及复分析的天然联系以及广泛的应用一直是泛函分析中活跃的研究方向。本项目将利用单位球Bergman空间上的平均值函数，单位球的划分，Bergman空间上积分和范数的估计，Bergman正交投影的性质，Bergman空间上的Balayage，向量值函数空间上的Schur’s测试，向量值Hilbert- Schmidt测试等这些函数论的工具和方法研究以下问题：1.刻画单位球Bergman空间上以局部可积函数为符号的Toeplitz算子有界和紧的充分必要条件;2.研究单位球Bergman空间上以无界函数为符号函数的Hankel算子和小Hankel算子的有界性和紧性等;3.在单位球向量值Bergman空间，刻画以矩阵值分段连续函数为符号函数的Toeplitz算子的紧性以及是Fredholm算子的充分必要条件并计算其Fredholm指标。

Operator theory in function space has been the active research direction in functional analysis since its application in various areas. In this project, we will study three questions by mean value function of the unit ball, integral and norm estimates on the Bergman space, the orthogonal projection on the Bergman space, Balayage, Schur’s test and vector valued Hilbert-Schmidt test in vector valued function space. Use these tools and methods we will study: 1.the necessary and sufficient conditions for bounded and compact Toeplitz operators with locally integrable functions; 2.the bounded properties and compact properties of the Hankel operators and small Hankel operators with unbounded symbol on Bergman space of the unit ball; 3.characterize the compactness of Toeplitz operators with piecewise continuous matrix valued function and the necessary and sufficient condition for Fredholm operators and calculate the Fredholm index.