Glöckner, Helge; Lucht, Lutz G. Weighted inversion of general Dirichlet series. (English) Zbl 1295.11095 Trans. Am. Math. Soc. 366, No. 6, 3275-3293 (2014). Let \(\Lambda\subseteq [0,+\infty)\) denote an additive semigroup, \(0\in\Lambda\). Let \(w: \Lambda\to[1,+\infty)\) be a weight function (satisfying a set of conditions given in the paper). Finally, let \(\mathcal{A}_w\) be the Banach algebra of functions \(a: \Lambda\to\mathbb{C}\) such that \[ \sum_{\lambda\in\Lambda} \left| a(\lambda)\right| w(\lambda) < +\infty, \] with standard linear operations and convolution defined by \( (a*b)(\lambda) = \sum_{\lambda_1+\lambda_2=\lambda} a(\lambda_1) b(\lambda_2). \) In particular, \(\mathcal{A}_1\supseteq \mathcal{A}_w\) is the algebra of absolutely summable complex functions on \(\Lambda\), corresponding to generalized Dirichlet series (supported on \(\Lambda\)) absolutely convergent in the half-plane \(\operatorname{Re}(s)\geq 0\). The authors investigate the group \(\mathcal{A}_w^*\) of invertible elements of \(\mathcal{A}_w\). They show that \[ \mathcal{A}_w^* = \{a\in\mathcal{A}_w : 0\notin \overline{\tilde a (\mathbb{H})}\},\tag{1} \] where \(\tilde a (\mathbb{H}) = \left\{\sum_{\lambda\in\Lambda} a(\lambda)e^{-\lambda s} : s\in \mathbb{C}, \operatorname{Re} s > 0\right\}\). In the case \(w=1\) this was previously obtained by D. A. Edwards [Proc. Am. Math. Soc. 8, 1067–1074 (1958; Zbl 0081.06805)].Two methods of proof are presented. The first relies on the theorem of Edwards. The authors show that \(\mathcal{A}_w^* = \mathcal{A}_w\cap \mathcal{A}_1^*\). They use, among others, a result from Gelfand’s theory, implying that \(a\in \mathcal{A}_w^*\) if and only if \[ \mathcal{A}_w^* = \{a\in\mathcal{A}_w : 0\notin \sigma(a)\},\tag{2} \] where \(\sigma(a)\) is the spectrum of \(a\), shown to be equal to \( \{h(a) : h\in\Delta(\mathcal{A}_w)\}, \) where \(\Delta(\mathcal{A}_w)\) is the set of non-trivial multiplicative linear functionals on \(\mathcal{A}_w\). The second method is independent of the earlier result and sheds more light on the similarity between (1) and (2). The authors are able to show that \(\tilde a (\mathbb{H})\) is dense in \(\sigma(a)\), and thus (2) implies (1).Some extensions of the main result and applications to multidimensional Dirichlet series are discussed in the final section. Reviewer: Maciej Radziejewski (Poznań) Cited in 2 Documents MSC: 11M41 Other Dirichlet series and zeta functions 30B50 Dirichlet series, exponential series and other series in one complex variable 30J99 Function theory on the disc 46H99 Topological algebras, normed rings and algebras, Banach algebras Keywords:general Dirichlet series; weighted inversion; Banach algebra; dual cone; rational vector space; separation theorem; Hahn-Banach theorem; rational polytope; semigroup algebra Citations:Zbl 0081.06805 PDFBibTeX XMLCite \textit{H. Glöckner} and \textit{L. G. Lucht}, Trans. Am. Math. Soc. 366, No. 6, 3275--3293 (2014; Zbl 1295.11095) Full Text: DOI arXiv References: [1] Christian Berg, Jens Peter Reus Christensen, and Paul Ressel, Harmonic analysis on semigroups, Graduate Texts in Mathematics, vol. 100, Springer-Verlag, New York, 1984. 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