The Ces\`{a}ro space of Dirichlet series and its multiplier algebra

TL;DR

This paper studies a Banach space of Dirichlet series with coefficients in Cesàro sequence spaces, analyzing its domain of analyticity, point evaluation boundedness, and identifying its multiplier algebra as a Wiener algebra shifted to a half-plane.

Contribution

It characterizes the space of Dirichlet series with Cesàro sequence coefficients and identifies its multiplier algebra as a shifted Wiener algebra, extending understanding of these function spaces.

Findings

Maximal domain of analyticity determined

Boundedness of point evaluations analyzed

Multiplier algebra identified as shifted Wiener algebra

Abstract

We consider the space $\mathcal{H}(ces_p)$ of all Dirichlet series whose coefficients belong to the Ces\`{a}ro sequence space $ces_p$, consisting of all complex sequences whose absolute Ces\`{a}ro means are in $\ell^p$, for $1<p<\infty$. It is a Banach space of analytic functions, for which we study the maximal domain of analyticity and the boundedness of point evaluations. We identify the algebra of analytic multipliers on $\mathcal{H}(ces_p)$ as the Wiener algebra of Dirichlet series shifted to the vertical half-plane $\mathbb{C}_{1/q}:=\{s\in\mathbb{C}:\Re s>1/q\}$, where $1/p+1/q=1$.