The Study Of Complex Symmetry,Boundedness And Compactness Of Composition Operators

Composition operators are a very important class of operators on various function spaces.In recent years,research of composition operators on holomorphic function spaces has been hot topics by many mathematical workers at home and abroad.The most fundamental research of composition operators is description of a series of properties such as boundedness,compactness,nature norm,spectrum,etc.on different function spaces.In this paper,we mainly investigate two parts:one is the complex symmetric of composition operators acting on the weighted Bergman space of unit disk;another is the description of two properties that are boundedness and compactness of composition operators on weighted ZygmundOrlicz spaces of unit disk.First of all,we discuss the complex symmetric of composition operators acting on the weighted Bergman space of unit disk.In Chapter 3,we will investigate the complex symmetric composition operators induced by linear fractional mappings,we had many results about this question on the Hardy space.Moreover,by studying we also find an example of the composition operators on the weighted Bergman space which is complex symmetric but not normal.Then,in Chapter 4 we will discuss boundedness and compactness of composition operators on weighted Zygmund-Orlicz spaces of unit disk.At first,using the definition of the Zygmund-Orlicz space,we construct the weighted Zygmund-Orlicz space.Secondly,we prove that each weighted Zygmund-Orlicz space is isometrically equal to certain weighted -Zygmund space for a special weight .Lastly,we restrict the Young's function,and describe boundedness and compactness of composition operators by using the equivalence of norms of two spaces.