Skip to main content

Algebras and Banach spaces of Dirichlet series with maximal Bohr’s strip

Revista Matemática Complutense (2022)Cite this article

Abstract

We study linear and algebraic structures in sets of Dirichlet series with maximal Bohr’s strip. More precisely, we consider a set \({\mathscr {M}}\) of Dirichlet series which are uniformly continuous on the right half plane and whose strip of uniform but not absolute convergence has maximal width, i.e., \(\nicefrac {1}{2}\). Considering the uniform norm, we show that \({\mathscr {M}}\) contains an isometric copy of \(\ell _1\) (except zero) and is strongly \(\aleph _0\)-algebrable. Also, there is a dense \(G_\delta \) set such that any of its elements generates a free algebra contained in \({\mathscr {M}}\cup \{0\}\). Furthermore, we investigate \(\mathscr {M}\) as a subset of the Hilbert space of Dirichlet series whose coefficients are square-summable. In this case, we prove that \({\mathscr {M}}\) contains an isometric copy of \(\ell _2\) (except zero).

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Alves, T.R., Carando, D.: Holomorphic functions with large cluster sets. Math. Nachr. 294, 1250–1261 (2021)

    MathSciNet  Article  Google Scholar 

  2. Aron, R.M., García, D., Maestre, M.: Linearity in non-linear problems. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 95, 7–12 (2001)

  3. Aron, R., Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Lineability: The search for linearity in Mathematics. Monographs and Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton (2016)

    MATH  Google Scholar 

  4. Aron, R.M., Bayart, F., Gauthier, P.M., Maestre, M., Nestoridis, V.: Dirichlet approximation and universal Dirichlet series. Proc. Am. Math. Soc. 145, 4449–4464 (2017)

    MathSciNet  Article  Google Scholar 

  5. Bartoszewicz, A., Glab, S.: Strong algebrability of sets of sequences of functions. Proc. Am. Math. Soc. 141, 827–835 (2013)

    MathSciNet  Article  Google Scholar 

  6. Bayart, F.: Linearity of sets of strange functions. Mich. Math. J. 53, 291–303 (2005)

    MathSciNet  Article  Google Scholar 

  7. Bayart, F.: Topological and algebraic genericity of divergence and of universality. Studia Math. 167, 161–181 (2005)

    MathSciNet  Article  Google Scholar 

  8. Bayart, F., Quarta, L.: Algebras in sets of queer functions. Isr. J. Math. 158, 285–296 (2007)

    MathSciNet  Article  Google Scholar 

  9. Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Linear subsets of nonlinear sets in topological vector spaces. Bull. Am. Math. Soc. 51, 71–130 (2014)

    MathSciNet  Article  Google Scholar 

  10. Bohnenblust, H.F., Hille, E.: On the absolute convergence of Dirichlet series. Ann. Math. (2) 32(3), 600–622 (1931)

    MathSciNet  Article  Google Scholar 

  11. Bohr, H.: Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichlet–Schen Reihen $\sum \frac{a_n}{n^s}$. Nachr. Ges. Wiss. Göttingen, Math. Phys. Kl. 441–488 (1913)

  12. Bohr, H.: Über die gleichmäßige Konvergenz Dirichletscher Reihen. J. Reine Angew. Math. 143, 203–211 (1913)

    MathSciNet  Article  Google Scholar 

  13. Conejero, J.A., Seoane-Sepúlveda, J.B., Sevilla-Peris, P.: Isomorphic copies of $\ell _1$ for $m$-homogeneous non-analytic Bohnenblust-Hille polynomials. Math. Nachr. 290(2–3), 218–225 (2017)

    MathSciNet  Article  Google Scholar 

  14. Defant, A., García, D., Maestre, M., Sevilla-Peris, P.: Dirichlet Series and Holomorphic Functions in High Dimensions (New Mathematical Monographs). Cambridge University Press, Cambridge (2019)

    Book  Google Scholar 

  15. Gurarij, V.I.: Linear spaces composed of non-differentiable functions. C. R. Acad. Bulg. Sci. 44(5), 13–16 (1991)

    MATH  Google Scholar 

  16. Queffélec, H., Queffélec, M.: Diophantine Approximation and Dirichlet Series. Harish-Chandra Research Institute Lecture Notes, vol. 2. Hindustan Book Agency, New Delhi (2013)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal do Amazonas, 69.077-000, Manaus, Brazil

    Thiago R. Alves & Leonardo Brito

  2. Departamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires and IMAS-UBA-CONICET, Buenos Aires, Argentina

    Daniel Carando

Authors
  1. Thiago R. Alves
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leonardo Brito
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel Carando
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Carando.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Thiago R. Alves was supported in part by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 and FAPEAM. Leonardo Brit was supported by FAPEAM. Daniel Carando was supported by CONICET-PIP 11220130100329CO, CONICET-PIP 11220200102366CO and ANPCyT PICT 2018-04104.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alves, T.R., Brito, L. & Carando, D. Algebras and Banach spaces of Dirichlet series with maximal Bohr’s strip. Rev Mat Complut (2022). https://doi.org/10.1007/s13163-022-00426-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13163-022-00426-1

Keywords

  • Dirichlet series
  • Lineability
  • Algebrability
  • Spaceability
  • Bohr’s strips

Mathematics Subject Classification

  • Primary 30B50
  • 46B87
  • Secondary 46E25
  • 30H50