On Compact Composition Operators On Some Holomorphic Function Spaces Over The Unit Ball

The paper gives some characterization theorems for the compact composition operators Cφon some function spaces over the unit ball Bn in Cn, which include BMOA space, Hardy space, weighted Bergman space, holomorphic Sobolev space, Besov space. Using the tools including the integral operator theory, Carleson mea-sure and pull-back measure for holomorphic mapping, we give the complete charac-terization for compact composition operators on these spaces.In the Introduction, we provide some background and knowledge for the prob-lems studied in the thesis. We mainly state the original and the development of the composition operator theory, finally we present the main results of our research.In Chapter one, we will give some preliminary knowledge for our research, including introduce some classic holomorphic function spaces and some theorems on these spaces.In Chapter two, main material is from a joint work with professor Song-Ying Li in [2]. We provide a characterization theorem for compact composition operators on BMOA(Bn), which generalize a result proved by Bourdon, Cima and Matheson for the case n=1 in [1].In Chapter three, a general function space X(Bn) over the unit ball in Cn with norm‖·‖X(Bn) is introduced. The space contains all Hardy space, Bergman space, Besov space etc.. We will gives a formulation of a compact composition operator on X(Bn). this work is main from the paper [3].In Chapter four, we summarize the present work of the paper, and make nota-tion for our future research.