The Boundedness And Compactness Of The Products Of Differentiation And Composition Operators

Operator theory in function spaces becomes a hot issue of research. As the carrier ofresearch is function spaces, the common operator must be derived from certain function,so we need to discuss the relation between the operator and the function. In this paper,we characterize the products of the diferentiation and composition operators actingfrom the bounded function space to the α Bloch space or Zygmund space in the ball,and necessary and sufcient conditions are given for the diference to be bounded andcompact.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 relatedto the article are given. They are the foundations and tools of the later work, and wewill use them without proof.The third chapter discusses the properties of the product operators, using thelemma and conclusions, the equivalent conditions of boundedness and compactness ofoperators are given.The fourth chapter,similar with the third, discuss another type of product opera-tors.At last, we summarize the work of the whole paper and give some open problemsabout the problem.