Difference Of Composition Operators On Bergman Spaces Over The Half-plane

This thesis is to study difference of composition operators on weighted Bergman spaces over half-plane Aαp.(Π＋)(0<p<∞,α>-1).Since II+ is a typical unbounded domain,the theory of composition operators on Aαp(Π+)is different from that on the Bergman space over the unit disk Aαp(D).This thesis is organized as follows.In chapter 1,we introduce the background and current progress of composition operators.In chapter 2,we introduce some results and notation on Aαp(Π+).In chapter 3,we consider that difference of composition operators Cφ-Cφ act-ing from Aαp(Π+)to general Lebesgue spaces Lq(p).In this chapter we overcome the unboundedness of half-plane and use Khinchine’s inequality and atom decomposition techniques to give joint Carleson measure characterizations of when such operator is bounded or compact.As applications,we obtain integral type characterizations of the bounded and compact difference of composition operators on such spaces and give concrete examples to show there exists bounded or compact difference of composition operators.In chapter 4,we consider double differences of composition operators a(Cφ1-Cφ2)-b(Cφ3-Cφ1),(Cφ1-Cφ2)-(Cφ3-Cφ4)on Aαp(Π+)where a,b ∈ C,a + b ≠0,ab≠0 and obtain complete characterizations for such operators.The relationship between angular derivatives and compact double differences of composition operators is also explored,which reveals some new phenomena due to the unboundedness of half-plane.In chapter 5,we give some joint Carleson measure characterizations of when the difference of composition operators is Hilbert-Schmidt from Aα2(Π+)to L2(μ).Futher-more,we give a sufficient condition when two composition operators Cφ and Cφ are in the same path component under operator norm.