Difference Of Composition Operators On The Spaces Of Holomorphic Functions

Operator theory on holomorphic function space as an important part of modern mathematics.Many mathematicians all over the world focus on the field of operator theory on holomorphic function spaces,which plays an important role in the devel-opment of functional analysis?differential geometry?von-Neumann algebra?dynamic system?quantum information,engineer control theory and other disciplines.In partic-ular,composition operators have been widely studied due to many interesting problems from operator theory and function theory can be modelled into some corresponding problems in the theory of composition operators.This thesis is to study the difference of composition operators on the spaces of holomorphic functions.This thesis is organized as follows:In Chapter 1,we introduce the background and current progress about the differ-ence of composition operators on the spaces of holomorphic functions.Based on the existing results,we explain the problems to be solved in this thesis.Chapter 2 is devoted to introduce the basic concepts and lemmas in this thesis,involving functional analysis,complex analysis,several variables complex analysis,real analysis and related contents of function spaces and operator theory.In Chapter 3,we investigate the difference of composition operators between the Hardy spaces on the unit disk by means of the Carleson measure.Compact difference of composition operators on Hardy spaces over the unit disk has been an open problem.We give an equivalent characterization for the compactness of the difference of composition operators on the Hardy spaces over the unit disk.Chapter 4 is aimed at studying the difference of generalized composition operators from the spaces of Cauchy integral transforms to other holomorphic function spaces.This is in sharp contrast with the classical case,here the characterizations about the dif-ference of generalized composition operators really do not involve the pseudo-hyperbolic metric between?(z)and?(z).Chapter 5 is concerned on the compact difference?order bounded difference and Hilbert-Schmidt difference of weighted composition operators between the weighted Bergman spaces on the unit ball.In addition,the complete characterizations for the compactness of double difference of composition operators on the weighted Bergman spaces over the unit ball are obtained in this chapter.In Chapter 6,we give some interesting characterizations about the compact differ-ence and Hilbert-Schmidt difference of composition operators on the weighted Bergman spaces over the unit disk.In Chapter 7,we investigate the difference of composition operators between the weighted Bergman spaces A_?~p(?~+)and the spaces B_?(?~+)?H_?~?(?~+)over the half-plane.Moreover,we give the new characterizations about the Hilbert-Schmidt difference of composition operators on the weighted Bergman spaces over the half-plane.