Fr\'echet spaces of general Dirichlet series

Abstract

Inspired by a recent article on Fr\'echet spaces of ordinary Dirichlet series $\sum a_n n^{-s}$ due to J.~Bonet, we study topological and geometrical properties of certain scales of Fr\'echet spaces of general Dirichlet spaces $\sum a_n e^{-\lambda_n s}$. More precisely, fixing a frequency $\lambda = (\lambda_n)$, we focus on the Fr\'echet 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-Fr\'echet 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 Fr\'echet 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.