# Banach spaces of general Dirichlet series

- Authors
- Publication Date
- Jul 04, 2018
- Source
- [email protected]
- Keywords
- License
- Unknown
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## Abstract

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 {lambda(n)} be a strictly increasing sequence of positive real numbers such that lim(n ->infinity) lambda(n) = infinity. We denote by H-infinity(lambda n) the complex normed space of all Dirichlet series D(s) = Sigma(n)b(n)lambda(-s)(n), which are convergent and bounded on the half plane [Re s > 0], endowed with the norm parallel to D parallel to(infinity) = sup( Re s>0) vertical bar D(s)vertical bar. If (*) there exists q > 0 such that inf(n), (lambda(q)(n+1) - lambda(q)(n)) > 0, then H-infinity (lambda(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, mu(n) = r(alpha) for n = p(alpha), and {lambda(n)} is the increasing rearrangement of the sequence {mu(n)}, then H-infinity (lambda(n)) is isometrically isomorphic to H-infinity (B-co). With this condition (*) we explain more explicitly the optimal cases of the difference among the abscissas sigma(c), sigma(b), sigma(u) and sigma(a). (C) 2018 Elsevier Inc. All rights reserved. / 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 {lambda(n)} be a strictly increasing sequence of positive real numbers such that lim(n ->infinity) lambda(n) = infinity. We denote by H-infinity(lambda n) the complex normed space of all Dirichlet series D(s) = Sigma(n)b(n)lambda(-s)(n), which are convergent and bounded on the half plane [Re s > 0], endowed with the norm parallel to D parallel to(infinity) = sup( Re s>0) vertical bar D(s)vertical bar. If (*) there exists q > 0 such that inf(n), (lambda(q)(n+1) - lambda(q)(n)) > 0, then H-infinity (lambda(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, mu(n) = r(alpha) for n = p(alpha), and {lambda(n)} is the increasing rearrangement of the sequence {mu(n)}, then H-infinity (lambda(n)) is isometrically isomorphic to H-infinity (B-co). With this condition (*) we explain more explicitly the optimal cases of the difference among the abscissas sigma(c), sigma(b), sigma(u) and sigma(a). (C) 2018 Elsevier Inc. All rights reserved. / 1