Banach spaces of general Dirichlet series

We study when the spaces of general Dirichlet series bounded on a half plane are Banach spaces, and show that some of those classes are isometrically isomorphic between themselves. In a precise way, let { λ n } be a strictly increasing sequence of positive real numbers such that lim n → ∞ ⁡ λ n = ∞ . We denote by H ∞ ( λ n ) the complex normed space of all Dirichlet series D ( s ) = ∑ n b n λ n − s , which are convergent and bounded on the half plane [ Re s > 0 ] , endowed with the norm ‖ D ‖ ∞ = sup Re s > 0 ⁡ | D ( s ) | . If (⁎) there exists q > 0 such that inf n ⁡ ( λ n + 1 q − λ n q ) > 0 , then H ∞ ( λ n ) is a Banach space. Further, if there exists a strictly increasing sequence { r n } of positive numbers such that the sequence { log ⁡ r n } is Q -linearly independent, μ n = r α for n = p α , and { λ n } is the increasing rearrangement of the sequence { μ n } , then H ∞ ( λ n ) is isometrically isomorphic to H ∞ ( B c 0 ) . With this condition (⁎) we explain more explicitly the optimal cases of the difference among the abscissas σ c , σ b , σ u and σ a .