Semigroups Of Composition Operators On Some Banach Spaces Of Analytic Functions

In this thesis we study several basic problems of composition semigroups on some classical Banach spaces of analytic functions on the unit disk,including the maximal subspaccs,strong continuity,spectra of infinitesimal generators of composition semigroups,and characterizations of composition semigroups.In Chapter 1,we introduce the background and the history of composition semigroups.Some related definitions and tools as well as main results of this thesis are presented.In Chapter 2,through studying the maximal subspaces of Qp generated by any non-trivial semigroup consisting of self-maps of the unit disk D,we provide the proper inclusion among Qp,Qp,0,and the maximal subspaces.As applications,our results answer a question proposed by A.G.Siskakis in 1996 and give an affirmative answer to a question in[20].Chapter 3 focuses on investigating spectra of infinitesimal generator of composition semigroups on weighted Bergman spaces.When the Denjoy-Wolff point of(?t)t?0 lies in D,we investigate the spectra of infinitesimal generators of composition semigroups on weighted Bergman spaces by studying some reg-ularity properties of composition semigroups.As byproducts,we prove some known results in different methods and obtain some results concerning about the boundedness,compactness as well as the spectrum of a certain integration operator.In Chapter 4,we characterize strong continuity of general operator semi-groups on Lebesgue spaces.In particular,a characterization of the strong conti-nuity of weighted composition semigroups on classical Hardy spaces and weight-ed Bergman spaces with regular weights is given.As applications,our result improves the results of A.G.Siskakis[81]and W.Konig[56]and answers a question of A.G.Siskakis proposed in[84].Meanwhile,we show that every composition semigroup is strongly continuous on A?p,1?p<?,provided ?satisfying a certain doubling property.In Chapter 5,several characterizations of strongly continuous operator semigroups on Dirichlet type spaces provided its infinitesimal generator of the form Af=Gf' are given;we also characterize strongly continuous semigroup-s of weighted composition operators on weighted Bergman spaces in terms of abelian intertwiners of multiplication operator Mz.