Vector-valued general Dirichlet series

Abstract

Opened up by early contributions due to, among others, H. Bohr, Hardy-Riesz, Bohnenblust-Hille, Neder and Landau the last 20 years show a substantial revival of systematic research on ordinary Dirichlet series a n n −s , and more recently even on general Dirichlet series a n e −λns . This involves the intertwining of classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. Motivated through this line of research the main goal of this article is to start a systematic study of a variety of fundamental aspects of vector-valued general Dirichlet series a n e −λns , so Dirichlet series, where the coefficients are not necessarily in C but in some arbitrary Banach space X.

Fréchet spaces of general Dirichlet series

“…for every n. Hence, applying (9) (for Q ε N ) and letting N → ∞ yields the claim in (8). In order to prove (ii), let us note that (8) immediately implies…”

“…One of our main tools is the representation of our pre-Fréchet spaces as countable projective limits of their natural 'Banach space precursors' (D ∞ (λ ) and H p (λ )). In this sense our article continues a series of recent articles on general Dirichlet series (see [8,11,14,26,25]), which combine classical results from the deep analysis presented by Hardy and Riesz in [17], with various topics from modern analysis (as complex analysis, functional analysis in Banach and Fréchet spaces, Fourier analysis on R, or har-monic analysis on compact abelian groups).…”

“…But in general (D ∞ (λ ), r ∞ ) is no Banach space, or equivalently it does not form a closed subspace of [26,Theorem 5.2]). The normed space D ∞ (λ ), and in particular the question when it is complete, was extensively studied in [8,11,14,26,25].…”

“…The basic definitions needed here are just straightforward translations of the scalar valued ones. The reader is referred to [8] for a complete account on the theory. In [12,Lemma 4.9] we have that, for any frequency λ , the mapping…”