Hostname: page-component-6bf8c574d5-b4m5d Total loading time: 0 Render date: 2025-02-16T19:09:00.410Z Has data issue: false hasContentIssue false
  • English
  • Français

The Fréchet Schwartz Algebra of Uniformly Convergent Dirichlet Series

Part of: Special classes of linear operators Topological linear spaces and related structures Holomorphic functions of several complex variables Series expansions

Published online by Cambridge University Press:  07 May 2018

José Bonet
José Bonet*
Affiliation:
Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València, E-46071 Valencia, Spain ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

The algebra of all Dirichlet series that are uniformly convergent in the half-plane of complex numbers with positive real part is investigated. When it is endowed with its natural locally convex topology, it is a non-nuclear Fréchet Schwartz space with basis. Moreover, it is a locally multiplicative algebra but not a Q-algebra. Composition operators on this space are also studied.

Keywords

Dirichlet seriesabscissas of convergenceFréchet spacescomposition operatorstopological algebras

MSC classification

Primary: 46A04: Locally convex Fréchet spaces and (DF)-spaces
Secondary: 30B50: Dirichlet series and other series expansions, exponential series 32A05: Power series, series of functions 46A03: General theory of locally convex spaces 46A11: Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) 46A45: Sequence spaces (including Köthe sequence spaces) 47B33: Composition operators
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 4 , November 2018 , pp. 933 - 942
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Apostol, T. M., Introduction to analytic number theory (Springer-Verlag, New York–Heidelberg, 1976).Google Scholar
2.Aron, R., Bayart, F., Gauthier, P. M., Maestre, M. and Nestoridis, V., Dirichlet approximation and universal Dirichlet series, Proc. Amer. Math. Soc. 145 (29107), 4449–4464.Google Scholar
3.Bayart, F., Opérateurs de composition sur les espaces de séries de Dirichlet et problèmes d'hypercyclicité simultanée, Thése de l'Université Lille 1, (2002).Google Scholar
4.Bayart, F., Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math. 136 (2002), 203–236.Google Scholar
5.Bayart, F., Compact composition operators on a Hilbert space of Dirichlet series, Illinois J. Math. 47 (2003), 725–743.Google Scholar
6.Bayart, F. and Mouze, A., Division et composition dans l'anneau des séries de Dirichlet analytiques, Ann. Inst. Fourier (Grenoble) 53 (2003), 2039–2060.Google Scholar
7.Bierstedt, K. D., Meise, R. G. and Summers, W. H., Köthe sets and Köthe sequence spaces, Functional Analysis, Holomorphy and Approximation Theory (ed. Barroso, J. A.), Volume 71, pp. 27–91 (North-Holland, Amsterdam, 1982).Google Scholar
8.Boas, H. P., The football player and the infinite series, Notices Amer. Math. Soc. 44 (1997), 1430–1435.Google Scholar
9.Bohr, H., Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletschen Reihen , Nachr. Ges. Wiss. Göttingen, Math. Phys. Kl. 1913 (1913), 441–488.Google Scholar
10.Bohr, H., Über die gleichmige Konvergenz Dirichletscher Reihen, J. Reine Angew. Math. 143 (1913), 203–211.Google Scholar
11.Defant, A., García, D. and Maestre, M., New strips of convergence for Dirichlet series, Publ. Mat. 54 (2010), 369–388.Google Scholar
12.Defant, A., García, D., Maestre, M. and Pérez-García, D., Bohr's strip for vector valued Dirichlet series, Math. Ann. 342(3) (2008), 533–555.Google Scholar
13.Defant, A., García, D., Maestre, M. and Sevilla-Peris, P., Bohr's strips for Dirichlet series in Banach spaces, Funct. Approx. Comment. Math. 44 (2011), 165–189.Google Scholar
14.Diestel, J., Sequences and series in Banach spaces (Springer, New York, 1984).Google Scholar
15.Dineen, S., Complex analysis on infinite-dimensional spaces (Springer, London, 1999).Google Scholar
16.Edwards, D. A., On absolutely convergent Dirichlet series, Proc. Amer. Math. Soc. 8 (1957), 1067–1074.Google Scholar
17.Finet, C. and Queffélec, H., Numerical range of composition operators on a Hilbert space of Dirichlet series, Linear Algebra Appl. 377 (2004), 1–10.Google Scholar
18.Gordon, J. and Hedenmalm, H., The composition operators on the space of Dirichlet series with square summable coefficients, Michigan Math. J. 46 (1999), 313–329.Google Scholar
19.Hedenmalm, H., Dirichlet series and functional analysis, In The legacy of Niels Henrik Abel (ed. Laudal, O. A. and Piene, R.), pp. 673–684 (Springer, Berlin, 2004).Google Scholar
20.Helson, H., Dirichlet series (Henry Helson, Berkeley, CA, 2005).Google Scholar
21.Hollstein, R., Generalized Hilbert spaces, Results Math. 8 (1985), 95–115.Google Scholar
22.Jameson, G. J. O., The prime number theorem (Cambridge University Press, Cambridge, 2003).Google Scholar
23.Jarchow, H., Locally convex spaces (B.G. Teubner, Stuttgart, 1981).Google Scholar
24.Lefévre, P., Essential norms of weighted composition operators on the space ℋ∞ of Dirichlet series, Studia Math. 191 (2009), 57–66.Google Scholar
25.Mallios, A., Topological algebras: selected topics, North-Holland Mathematics Studies, Volume 124 (North-Holland, Amsterdam, 1986).Google Scholar
26.Maurizi, B. and Queffélec, H., Some remarks on the algebra of bounded Dirichlet series, J. Fourier Anal. Appl. 16 (2010), 676–692.Google Scholar
27.Meise, R. and Vogt, D., Introduction to functional analysis (The Clarendon Press, New York, 1997).Google Scholar
28.Michael, E., Locally multiplicatively-convex topological algebras, Mem. Amer. Math. Soc. 11 (1952), 79 pages.Google Scholar
29.Queffélec, H., Bohr's vision of ordinary Dirichlet series: old and new results, J. Anal. 3 (1995), 43–60.Google Scholar
30.Queffélec, H., Espaces de séries de Dirichlet et leurs opérateurs de composition, Ann. Math. Blaise Pascal 22 (2015), 267–344.Google Scholar
31.Queffélec, H. and Queffélec, M., Diophantine approximation and Dirichlet series (Hindustain Book Agency, New Delhi, 2013).Google Scholar
32.Wang, M. and Yao, X., Topological structure of the space of composition operators on ℋ∞ of Dirichlet series, Arch. Math. (Basel) 106 (2016), 471–483.Google Scholar
33.Żelazko, W., Selected topics in topological algebras, Lectures 1969/1970. Lecture Notes Series, Volume 31 (Matematisk Institut, Aarhus Universitet, Aarhus, 1971).Google Scholar