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

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).

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Correspondence to Daniel Carando.

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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.

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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

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  • 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