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 {λn} be a strictly increasing sequence of positive real numbers such that limn→∞⁡λn=∞. We denote by H∞(λn) the complex normed space of all Dirichlet series D(s)=∑nbnλn−s, which are convergent and bounded on the half plane [Res>0], endowed with the norm‖D‖∞=supRes>0⁡|D(s)|. If (⁎) there exists q>0 such that infn⁡(λn+1q−λnq)>0, then H∞(λn) is a Banach space. Further, if there exists a strictly increasing sequence {rn} of positive numbers such that the sequence {log⁡rn} is Q-linearly independent, μn=rα for n=pα, and {λn} is the increasing rearrangement of the sequence {μn}, then H∞(λn) is isometrically isomorphic to H∞(Bc0). With this condition (⁎) we explain more explicitly the optimal cases of the difference among the abscissas σc,σb,σu and σa.