Fréchet spaces of general Dirichlet series

Abstract

Inspired by a recent article on Fréchet spaces of ordinary Dirichlet series ∑ a n n −s due to J. Bonet, we study topological and geometrical properties of certain scales of Fréchet spaces of general Dirichlet spaces ∑ a n e −λ n s . More precisely, fixing a frequency λ = (λ n ), we focus on the Fréchet space of λ -Dirichlet series 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 a n e −λ n /k for each k equals the Fourier coefficients of a function on an appropriate λ -Dirichlet group.