Spaces of Dirichlet Series

This work is dedicated to the study of multiple Dirichlet series and it focuses on three main aspects: convergence, spaces of bounded multiple Dirichlet series and the composition operators of such spaces. In the first chapter we give the necessary preliminary results on regular convergence of double and multiple series and its equivalence with the definition of convergence in a restricted sense. We also recall sequences of bounded variation and we show how they are the multipliers of convergent series in the space of sequences, extending also this characterization to double and multiple regularly conergent series. In the second chapter we recall the fundamentals on the theory of bounded ordinary Dirichlet series of one complex variable. Frist we discuss the different half-planes of convergence and their formulae, reviewing different formulae also for general Dirichlet series. Then we give Bohr's fundamental result which identifies the abscissa of uniform convergence and the abscissa of boundedness, providing a proof of a quantitative version of this result which is key to prove that the space of Dirichlet series that are convergent and bounded form a Banach algebra. Chapter three is dedicated to obtaining theorems on regular convergence of multiple Dirichlet series, and also the study of absolute convergence and uniform convergence, since it is a remarkable fact in the theory of Dirichlet series that different types of convergence produce different regions of convergence. We study sets of regular convergence following the previous work of Kojima, providing a new proof or his characterization of such sets, and we give new formulae for obtaining such sets in the double ordinary case. With this formulae we are able to obtain non-trivial examples of sets of regular convergence. In the fourth chapter we set the ground of the theory of bounded multiple Dirichlet series. We study first the double case: we define the space of bounded double Dirichlet series and, through a vector-valued perspective, we show that this space is a Banach algebra. The key result here is Bohr's Theorem, both in its scalar version and in its vector-valued version, so the first aim in the more general multiple case is to obtain an multiple analogue for this result. Once we have done that, we can show that the spaces of bounded multiple Dirichlet series are Banach algebras, and moreover we show that they are all isometrically isomorphic, independently of the dimension. In the fifth and last chapter we study composition operators on spaces of double Dirichlet series. First we review the characterization of composition operators of the space of bounded Dirichlet series of one complex variable, and we improve such characterization by dropping the hypothesis of holomorphy of the symbol. Then we focus our interest on the characterization of the composition operators of the space of bounded double Dirichlet series. We also show how the composition operators of this space of Dirichlet series are related to the composition operators of the corresponding spaces of holomorphic functions. Finally, we give a characterization of the superposition operators in the space o bounded Dirichlet series and the Hardy spaces of Dirichlet series.