Composition Operators On Spaces Of Dirichlet Series

The study of this thesis is mainly on the theory of composition operators on some spaces of Dirichlet series, involving invariant subspaces, cyclic phenomenons, topolog-ical structures etc. We also consider the complex symmetry of weighted composition operators on the reproducing kernel function spaces over several variables.In Chapter 1, we introduce the background and current progress of some spaces of analytic functions, i.e., some related notation and results about composition operators on those spaces.Chapter 2 is devoted to studying invariant subspaces of composition operators on the Hilbert space H of Dirichlet series with square summable coefficients. Particularly, the strongly closed unital algebras generated by some composition operators are shown to be reflexive. As an application, we provide a criterion for composition operators with certain symbols not to be algebraic.In Chapter 3, we study weighted composition operators on the Hilbert space H. Especially, the Hermitianness, Fredholmness and invertibility of such operators are char-acterized, and the spectra of compact and invertible weighted composition operators are also described.Chapter 4 is aimed at investigating topological structures of the space of composi-tion operators acting on the space H? of Dirichlet series. Unexpectedly, we show that there are two compact composition operators which are not in the same path component on H?. This is in sharp contrast with the classical case where all compact composition operators on H? of one variable or several variables lie in the same path component.Chapter 5 is concerned on some properties of composition operators on Hilbert spaces H(E,?s) of entire Dirichlet series, which include the Fredholmness, Hilbert-Schmidtness, spectra, cyclic and hypercyclic phenomenons, and also answer a norm question raised by Cowen-MacCluer.In Chapter 6, we investigate complex symmetry of weighted composition operators on the reproducing kernel function spaces Hs (s>0) over several variables. First, we give explicit forms of complex symmetric weighted composition operators with respect to a specific conjugation on Hs. Moreover, we characterize completely the compactness, normality and isometry of such operators, and estimate the spectral radii for such operators.