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).