Composition Operator Theory On Spaces Of Dirichlet Series

As important parts in operator theory,composition operators on holomorphic function spaces have been widespread concern by a growing number of mathematicians all over the world due to the fact that they are inextricably liked with C*-algebra,Calkin algebra,operator semigroups and other topics.In classical function spaces,the basic questions of operators such as boundedness,compactness,topological structure are studied based on the methods and theories of holomorphic function theory.GordonHedenmalm first characterized the boundedness of composition operators acting on the Hardy-Hilbert space of Dirichlet series.Since then,the investigation of composition operators on function spaces of Dirichlet series has become a hot topic.This thesis mainly describes the boundeness and compactness of composition operators on Banach spaces of Dirichlet series.This thesis is organized as follows.In Chapter 1,we review some background and development of composition operators on the spaces of holomorphic functions.In addition,based on the research achievement available in literature,we demonstrate the motivations of our topic and explain the problems to be solved in this thesis.Chapter 2 is devoted to discussing several basic notions and properties associated to abscissas of convergence,vertical limit functions,some function spaces and operator theory which are necessary for the subsequent research.In Chapter 3,we introduce generalized Nevanlinna counting functions and restricted counting functions in the weighted Bergman spaces of Dirichlet series.Moreover,some estimates of generalized Nevanlinna counting functions and restricted counting functions are given by using the Littlewood inequality of Nevanlinna counting function on the unit disk and conformal mapping.Chapter 4 is aimed at investigating the boundedness of composition operators induced by Dirichlet series symbols on the weighted Bergman spaces of Dirichlet series.Moreover,we establish the Littlewood-Paley formulas on the weighted Bergman spaces of Dirichlet series.Using these results and some estimates about generalized restricted Nevanlinna counting function,we present some sufficient conditions for compactness of composition operators on the weighted Bergman spaces of Dirichlet series.Furthermore,we obtain some necessary conditions for compactness of composition operators on these spaces in terms of conjugate operators.Chapter 5 gives some new characterizations on Carleson measures for Bohr-Bergman spaces of Dirichlet series by using the connections of the non-conformal Bergman spaces and Bohr-Bergman spaces of Dirichlet series.Moreover,we study the boundedness and compactness of composition operators from Bohr-Bergman spaces to Hardy-Hilbert space of Dirichlet series in terms of Carleson measures and the Littlewood-Paley formula on Hardy-Hilbert space of Dirichlet series.Furthermore,we completely characterize the boundedness and compactness of composition operators from Bohr-Bergman spaces to standard weight Bergman spaces of Dirichlet series.