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

TL;DR

This paper explores the algebraic and geometric structures of sets of Dirichlet series with maximal Bohr's strip, revealing rich linear, algebraic, and topological properties in different functional spaces.

Contribution

It establishes the presence of isometric copies of  and  in the set of Dirichlet series with maximal Bohr's strip and demonstrates strong algebraic structures such as -algebrability and free algebra generation.

Findings

Contains an isometric copy of  in the set of Dirichlet series

Contains an isometric copy of  in the Hilbert space setting

Exhibits strong algebraic structures including -algebrability and free algebra

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., $\frac{1}{2}$. 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).