The Boundedness And Compactness Of The Weighted Composition Operator

As a question which takes more attentions of the academia in recent years, the algebraic structure of weighted composition operator is the difficult content when s-tudying it. So it is very significant for us to study the boundedness, compactness and essential norm of the weighted composition operators between different spaces of the holomorphic functions. This paper also studies the boundedness, compactness and es-sential norm of the weighted composition operators in some spaces of the holomorphic functions and introduces some new or equivalent estimates by analysis.This paper is designed to be divided into four parts to introduce the above questions in detail.The first part is mainly about the introduction of the background of this branch and the process we study the weighted composition operators.The second chapter introduces some priority knowledge which includes some basic notations, definitions and will be used in the following parts of this paper.The third chapter mainly gives the new estimates of the boundedness and essential norm of the weighted composition operator uC?:Z?? B? in the unit disk.At last, we show the boundedness of compactness of the weighted composition operator uC?:B and Bp?Z in the unit disk and give some equivalent conditions.