Multipliers of the Hilbert spaces of Dirichlet series

TL;DR

This paper characterizes the multiplier algebra of certain Hilbert spaces of Dirichlet series defined by positive sequences, linking it to bounded holomorphic functions and extending previous classifications.

Contribution

It provides a new isometric isomorphism for the multiplier algebra under specific conditions on the weight sequence, generalizing prior results by Stetler.

Findings

Multiplier algebra is isometrically isomorphic to bounded holomorphic functions under conditions.

Established conditions on weight sequences for the multiplier algebra.

Extended Stetler's classification to broader classes of weights.

Abstract

For a sequence $\mathbf w = \{w_j\}_{j = 2}^\infty$ of positive real numbers, consider the positive semi-definite kernel $\kappa_{\mathbf w}(s, u) = \sum_{j = 2}^\infty w_j j^{-s - \overline{u}}$ defined on some right-half plane $\mathbb H_{\rho}$ for a real number $\rho.$ Let $\mathscr H_{\mathbf w}$ denote the reproducing kernel Hilbert space associated with $\kappa_\mathbf w.$ Let \begin{equation*} \delta_{\mathbf w} = \inf\Bigg\{\Re(s) : \sum\limits_{\substack{j \geqslant 2 \\ \tiny{\textbf{gpf}}(j) \leqslant p_n }} w_j j^{- s} < \infty ~\text{for all}~ n \in \mathbb Z_+\Bigg\}, \end{equation*} where $\{p_j\}_{j \geqslant 1}$ is an increasing enumeration of prime numbers and $\textbf{gpf}(n)$ denotes the greatest prime factor of an integer $n \geqslant 2.$ If $\mathbf w$ satisfies \begin{equation*} \sum_{\substack{j \geqslant 2\\ j | n}} j^{-\delta_\mathbf w} w_j \mu\Big(\frac{n}{j}\Big) \geqslant 0,\quad n \geqslant 2, \end{equation*} where $\mu$ is the M$\ddot{\mbox{o}}$bius function, then the multiplier algebra $\mathcal M(\mathscr H_\mathbf w)$ of $\mathscr H_\mathbf w$ is isometrically isomorphic to the space of all bounded and holomorphic functions on $\mathbb H_\frac{\delta_{\mathbf w}}{2}$ that are representable by a convergent Dirichlet series in some right half plane. As a consequence, we describe the multiplier algebra $\mathcal M(\mathscr H_\mathbf w)$ when $\mathbf w$ is an additive function satisfying $\delta_{\mathbf w} \leqslant 0$ and \begin{align*} \frac{w_{p^{j-1}}}{w_{p^j}} \leqslant p^{-\delta_{\mathbf w}}~\text{for all integers} ~~ j \geqslant 2~\mbox{and all prime numbers}~p. \end{align*} Moreover, we recover a result of Stetler that classifies the multipliers of $\mathscr H_\mathbf w$ when $\mathbf w$ is multiplicative. The proof of the main result is a refinement of the techniques of Stetler.