Fréchet spaces of general Dirichlet series

Abstract

Inspired by a recent article on Fréchet spaces of ordinary Dirichlet series ∑ ann-s due to J. Bonet, we study topological and geometrical properties of certain scales of Fréchet spaces of general Dirichlet spaces ∑ane-λns focus on the Fréchet space of λ-Dirichlet series ∑ane-λns which have limit functions bounded on all half planes strictly smaller than the right half plane [Re>0]. We develop an abstract setting of pre-Fréchet spaces of λ-Dirichlet series generated by certain admissible normed spaces of λ-Dirichlet series and the abscissas of convergence they generate, which allows also to define Fréchet spaces of λ-Dirichlet series for which ane-λn/k for each k equals the Fourier coefficients of a function on an appropriate λ-Dirichlet group.