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


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).

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

Mathematics Subject Classification

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