Some Operators On The Zygmund Space

In this dissertation,we mainly study the boundedness and compactness of generalized composition operator,pointwise multiplier and weighted Ces(?)ro operator between Zygmund spaces and other analytic function spaces.To study the boundedness and compactness of these operators,we mainly use their definitions and some properties of the norm,and select the appropriate auxiliary function or test function,to find the conditions that the holomorphic function u and holomorphic self-mapφsatisfied.Thus,we obtain necessary and sufficient conditions that these operators are bounded operators or compact operators in different holomorphic function spaces.The study of generalized composition operator,pointwise multiplier and weighted Ces(?)ro operator links analytic function theory and operator theory.Its main goal is to use some results and methods of classical analytic function theory to exploit some of the most basic questions of which can be asked about linear operators and function spaces. At the same time,using operator theory as a tool determines the classical questions in function theory.These operators have intensively studied by many authors and a lot of profound results have been obtained,but there are still a number of interesting questions about these operators unsolved.We believe that these operators theory must be living as the mathematicians' efforts and the publication of these books.This article consists of four chapters as follows:In the first chapter,we briefly describe the research background,procession of development, the value and significance of the generalized composition operator,pointwise multiplier and weighted Ces(?)ro operator.And make a simple introduction of the function spaces we will used in the back of the chapter.In the second chapter,we introduce the boundedness and compactness of generalized composition operators C_φ^h from the weighted Bergman spaces to the Zygmund spaces as well as the little Zygmund spaces.It is obtained as follows:(1) the sufficient and necessary condition for C_φ^h to be bounded or compacted operators from A_α^p to Z; (2) the sufficient and necessary condition for C_φ^h to be bounded or compacted operators from A_α^p to Z₀.In the third chapter,this paper discusses the pointwise multipliers from Zygmund type spaces Z^p to Z^q and the pointwise multipliers from the general function spaces F(p,q,s) to Zygmund type spaces in the unit ball B of Cⁿ.It is obtained as follows: (1)the sufficient and necessary condition forψ∈M(Z^p,Z^q);(2)the sufficient and necessary condition forψ∈M(Z₀^p,Z₀^q);(3)the sufficient and necessary condition forψ∈M(F(p,q,s),Z^δ).In the fourth chapter,we discuss the boundedness and compactness of the weighted Ces(?)ro operators T_g on the weighted Zygmund spaces Z^p in the unit ball of Cⁿ as well as the little Zygmund spaces Z₀^p in the unit ball of Cⁿ.It is obtained as follows:(1)the sufficient and necessary conditions for T_g to be bounded or compacted operators from Z^p to Z^q;(2)the sufficient and necessary conditions for T_g to be bounded or compactedoperators from Z₀^p to Z₀^q.