Banach spaces of general Dirichlet series
type: Item
instance of: scholarly article
title: Banach spaces of general Dirichlet series
zbMATH Open document ID: 1408.30001
author: Yun Sung Choi
author: Un Young Kim
author: Manuel Maestre
published in: Journal of Mathematical Analysis and Applications
publication date: 2018-07-04T00:00:00Z
review text: Fix a strictly increasing sequence \(\lambda = (\lambda_n)\) of positive real numbers which tends to infinity. The authors study general Dirichlet series of the form \(D=\sum a_n \lambda_n^s\), where the coefficients \((a_n)\) form a sequence of complex numbers, and \(s\) is a complex variable. Denote by \(\mathcal{H}_\infty(\lambda)\) the space of all such series which converge on the positive right half space \([\text{Re} >0]\) and which define there a bounded (and then necessarily holomorphic) function. The main interest of this article is to show that under the condition [\(\exists q >0: \inf (\lambda_{n+1}^q - \lambda_n^q )] >0\)], the space \(\mathcal{H}_\infty(\lambda)\) together with the norm \(\|D\|_\infty = \sup_{\text{Re}s>0} |\sum a_n \lambda_n^s| \) forms a Banach space. This generalizes the well-known case of ordinary Dirichlet series \(\sum a_n n^{-s}\). The authors observe that their condition on \(\lambda\) in fact is equivalent to a well-known condition of Bohr [\(\exists \ell \geq 0 \;\forall \delta > 0: (\log \lambda_{n+1} /\lambda_n )^{-1} = O (\lambda_n^{\ell + \delta})\)]. \textit{H. Bohr} [J. Reine Angew. Math. 143, 203--211 (1913; JFM 44.0307.01)] showed that under his condition on \(\lambda\) for each \(D=\sum a_n \lambda_n^s\) which converges at least in some \(s\), the following two facts hold true: (1) \(D\) converges absolutely on some appropriate half space. (2) If the function defined by \(D\) extends to a bounded and holomorphic function on \([\text{Re} > 0]\), then \(D\) converges uniformly on each smaller half space \([\text{Re} > \varepsilon]\). In the spirit of Bohr's approach the authors provide another proof of these two facts which then in an almost straightforward way lead to the desired completeness of \(\mathcal{H}_\infty(\lambda)\). Additionally, it is shown that under certain (quite restrictive) further conditions on \(\lambda\) the Banach space \(\mathcal{H}_\infty(\lambda)\) can be isometrically identified with the Bana…
reviewed by: Andreas Defant
Mathematics Subject Classification ID: 30B50
zbMATH DE Number: 6898333
zbMATH Keywords: Dirichlet series
zbMATH Keywords: abscissas of convergence
zbMATH Keywords: holomorphic functions
MaRDI profile type: Publication
full work available at URL: j.jmaa.2018.05.036
OpenAlex ID: W2802435927
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Wikidata QID: Q129835129
DOI: 10.1016/J.JMAA.2018.05.036
description: scientific article; zbMATH DE number 6898333