Research On Complex Symmetry And Difference Of Composition Operators

Currently,the operator theory in function space is a hot issue studied by many scholars,it discusses the properties of some operators in the function space,includ-ing complex symmetry,boundedness,compactness,spectrum and nature-norm.The complex symmetry of the composition operators on the Bergman space induced by the holomorphic self-map of the unit disk D and the compactness and boundedness of dif-ferences between products of composition operators and integral-type operators on the logarithmic Bloch-type space are investigated.This article is divided into six chapters:the first chapter is introduction section gives the background of the composition operators,research status of composition op-erators and some main results of this article.The second chapter is the preliminary knowledge,which detailed introduction the composition operators,the complex sym-metry operator,the integral-type operators,the intertwining relationship and some con-cepts and properties of several Banach spaces involved in this paper.The third chapter introduces the complex symmetry of the composition operators on the Bergman space A~2(D)induced by the holomorphic self-map of the unit disk D.The fourth chapter gives an example of a symmetry operator that is not a normal operator on A~2(D).The fifth chapter introduces the intertwining relations for composition operators and integral-type operators on the logarithmic Bloch-type space.The sixth chapter is the concluding remarks,which puts some unsolved problems in this paper.The innovation of this paper is to generalize the conclusion of the complex symmetry of the composition operators in Hardy space to Bergman space,and find an instance of Bergman space which is not a normal operator,but a complex symmetric operator.Secondly,this paper extends the intertwining relations between composition operators and integral-type operators to the Logarithmic Bloch-type space.