Weighted Composition Operators On Several Holomorphic Function Spaces

This dissertation collects the main results obtained by the author during the period when she has applied for the M.D.Study of composition operators is the product of combination of the analytic function theory and operator theory. Early in the sixties and seventies of last century, people have concerned about the problems of the nature and application of composition operators. Then, they were generalized to the weighted composition operators, this is a very important specific operator category. In recent years, research on such operators has aroused the interest of many workers in math at home and abroad. So far, has acquired a much more profound conclusions. In this paper, on the basis of the results which our predecessors have done, we will further discuss the nature of composition operators and weighted composition operators. We also improve many results of some literature references , and extend to more function spaces. Make the nature of the operators more perfect.In this article, we focused on the two nature of the composition operators and weighted composition operators on several holomorphic function spaces on the unit disc: boundedness and compactness. Mainly apply the definition of operators and the related nature of norm. And select the appropriate auxiliary function or test function, find the conditions that the holomorphic function u and holomorphic self-mapφshould satisfy. Thus we obtain necessary and sufficient conditions that the composition operators and weighted composition operators are bounded operators or compact operators in different holomorphic function spaces.This article is divided into three chapters. One of the first chapter, we briefly describes the research background, development process, research value and significance of the composition operators. And make a simple introduction of the function spaces we will used in the back of the chapter. In the second chapter, we discuss the boundedness of the composition operators and weighted composition operators. And received the sufficient and necessary conditions.In the third chapter, on the basis of the results in the second chapter, we discuss the compactness of the composition operators and weighted composition operators. And received the sufficient and necessary conditions.