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 $$\sum a_n e^{-\lambda _n s}$$ focus on the Fréchet space of $$\lambda $$ -Dirichlet series $$\sum a_n e^{-\lambda _n s}$$ 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.