Hardy space of translated Dirichlet series

Abstract

[EN] We study the Hardy space of translated Dirichlet series H+. It consists on those Dirichlet series Sigma a(n)n(-s) such that for some (equivalently, every) 1 <= p < infinity, the translation Sigma a(n)n(-(s+ 1/sigma)) belongs to the Hardy space H-p for every sigma > 0. We prove that this set, endowed with the topology induced by the seminorms {parallel to center dot parallel to(2,k)}(k is an element of N) (where parallel to Sigma a(n)n(-s) parallel to(2,k) is defined as parallel to Sigma a(n)n(-(s+ 1/k))parallel to(H2)), is a Frechet space which is Schwartz and non nuclear. Moreover, the Dirichlet monomials {n(-s)}(n is an element of N) are an unconditional Schauder basis of H+. We also explore the connection of this new space with spaces of holomorphic functions on infinite-dimensional spaces. In the spirit of Gordon and Hedenmalm's work, we completely characterize the composition operator on the Hardy space of translated Dirichlet series. Moreover, we study the superposition operators on H+ and show that every polynomial defines an operator of this kind. We present certain sufficient conditions on the coefficients of an entire function to define a superposition operator. Relying on number theory techniques we exhibit some examples which do not provide superposition operators. We finally look at the action of the differentiation and integration operators on these spaces.