Algebraic Properties Of Composition Operators

Algebraic properties of composition operators in analytic function spaces are im-portant component parts in operator fields. In this paper, we characterize essential commutativity of the integral operators and composition operators, the boundedness and compactness of the weighted differentiation composition operator, and some asymp-totically equivalent expressions of the essential norms of difference of products of the integral type and composition operators in some analytic function spaces.The paper is mainly divided into five chapters.The first chapter is the introduction of the whole paper. We talk about the back-ground of this paper, and make plans for the research of the problem.Next chapter of the paper, some important concepts and lemmas which are related to the article are given. They are the foundations and tools of the later work, and we will use them without proof.The third chapter discusses the properties of essential commutativity of the integral operators and composition operators from the mixed-norm space to the Bα space in the unit disk.The fourth chapter characterizes the boundedness and compactness of the weighted differentiation composition operator from weighted Bergman space to n-th weighted space on the unit disk of C.At last, we characterizes the boundedness of the differences of the products of integral operators and composition operators from the mixed-norm space to μ-Bloch type space on the unit ball of Cn. At the same time, some asymptotically equivalent expressions of the essential norms of difference of products of integral operators and composition operators are presented.