Fréchet spaces of general Dirichlet series

Abstract

Inspired by a recent article on Fréchet spaces of ordinary Dirichlet series $\sum a_n{n}^{-s}$ due to J.~Bonet, we study topological and geometrical properties of certain scales of Fréchet spaces of general Dirichlet spaces $\sum a_n{e}^{-{\lambda }_{n}s}$. More precisely, fixing a frequency $\lambda =({\lambda }_n)$, we focus on the Fréchet space of $\lambda$-Dirichlet series which have limit functions bounded on all half planes strictly smaller than the right half plane $[\mathrm{Re}>0]$. We develop an abstract setting of pre-Fréchet spaces of $\lambda$-Dirichlet series generated by certain admissible normed spaces of $\lambda$-Dirichlet series and the abscissas of convergence they generate, which allows also to define Fréchet spaces of $\lambda$-Dirichlet series for which $a_n{e}^{-{\lambda }_n/k}$ for each $k$ equals the Fourier coefficients of a function on an appropriate $\lambda$-Dirichlet group.