Products Of Toeplitz Operators With Quasihomogeneous Symbols On The Bergman Space

This paper concerns with the product of Toeplitz operators on the Bergman space with symbols in the quasihomogeneous functions.Three cases in terms of the degree of the symbol are discussed.By using Mellin transform,the necessary and sufficient conditions are given for the product of two Toeplitz operators to be a Toeplitz operator.Moreover,we give an explicit formula and an example for the symbol of the product in certain cases.These results improve the conclusions in [1].This thesis consists of four chapters as follows:In Chapter 1,the progress of the research in the product of Toeplitz operators is introduced.Chapter 2 is concerned with the basic structure of Bergman space and the definition of Toeplitz operator on the Bergman space.In chapter 3,we introduce the Mellin transform,the most useful tools in the article,and discuss the properties of the Mellin transform,the Mellin convolution and Toeplitz operators with symbols in quasihomogeneous functions.Chapter 4 is the main part of this paper.Three cases in terms of the degree of the symbol are discussed.The necessary and sufficient conditions are given for the product of two Toeplitz operators to be a Toeplitz operator.Moreover,we give an explicit formula for the symbol of the product in certain cases.