Weighted Composition Operators On The Hardy Space

The title of this article is Weighted Composition Operators on the Hardy Spaces. That is to say, by using the tool of the essential norm, we get the upper or lower estimates of the weighted composition operators which we have studied. In addition, we get a sufficient or necessary condition for the weighted composition operator to be compact. The article is divided into four parts.In the first part, some results and methods about operator theory in function spaces of several complex variables, such as composition operators, weighted composition operators and multipliers, are introduced briefly .It can be taken as a preliminary.In next part, some important concepts and theorems which are related to the article are given.The third part contains some lemmas and their proofs which are necessary to the proofs of our main theorems.The last part contains some preparatory lemmas, and proofs of the main theorems in the article.