Boundedness And Compactness Of Composition Operators On Bergman Spaces

As classic operators in functional analysis,composition operators on the analytic function spaces have gained increasing interest from operator theory.In this paper,we consider the boundedness and compactness of composition operators on Bergman spaces induced by analytic mappings over distinct domains.In Chapter 1,we introduce the background and research progress of the boundedness and compactness of composition operators on Bergman spaces in complex plane and Bergman spaces in several variables.In Chapter 2,we consider composition operators defined between distinct Bergman spaces over planar domains.The smoothness on boundaries of the domains plays an important role in our study.On one hand,an essential extension of Littlewood’s Subordination Principle is obtained.Precisely,for each analytic mapping that is defined between bounded domains of smooth boundaries,the associated composition operator is always bounded.This essentially depends on a standard decomposition of analytic mappings over a classical domain(bounded by finitely many disjoint circles).On the other hand,the situation becomes complex if domains with cusp boundary points are concerned,and there exists a link between the boundary behavior of the function symbol and the boundedness of the associated composition operator,where a detailed discussion is presented.Finally,we give estimates of norms for some classes of such composition operators.In Chapter 3,we elaborate on the compactness of composition operators defined between distinct Bergman spaces over planar domains.Firstly,MacCluer and Shapiro’s result[Canad.J.Math.,38:878-906,1986]on compactness of composition operators have been widely generalized to domains with smooth boundaries.Secondly,we construct a counter-example to show that if the smoothness of boundary is removed,the result does not hold.Finally,some cases are discussed for domains with cusp points.In Chapter 4,we study the proper analytic mapping between distinct domains in several variables.We give a necessary condition for the boundedness of the composition operator on the associated Bergman space.For some domains with smooth boundaries,such as polydisks and unit balls,we construct a class of unbounded composition operators on Bergman spaces induced by proper mappings.In Chapter 5,let A be a unital algebra and M be a unital A-bimodule.We characterize the linear mappings δ and τ from A into M,satisfying δ(A)B+Aτ(B)=0 for every A,B ∈ A with AB=0 when A contains a separating ideal T of M,which is in the algebra generated by all idempotents in A.We apply this result to P-subspace lattice algebras,completely distributive commutative subspace lattice algebras,and unital standard operator algebras.Furthermore,suppose that A is a unital Banach algebra and M is a unital Banach A-bimodule,we give a complete description of linear mappings δ and τ from A into M,satisfying δ(A)B+Aτ(B)=0 for every A,B∈A with AB=I.