Multipliers of the Hilbert spaces of Dirichlet series

Chaman Kumar Sahu

Published 2023-01-16Version 1

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 \geq 2 \\ \textbf{gpf}(j) \leq p_n }} w_j j^{- s} < \infty ~\text{for all}~ n \in \mathbb Z_+\Bigg\}, \end{equation*} where $\{p_j\}_{j \geq 1}$ is an increasing enumeration of prime numbers and $\textbf{gpf}(n)$ denotes the greatest prime factor of an integer $n \geq 2.$ If $\mathbf w$ satisfies \begin{equation*} \sum_{\substack{j \geq 2\\ j | n}} j^{-\delta_\mathbf w} w_j \mu\Big(\frac{n}{j}\Big) \geq 0,\quad n \geq 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 $\delta_{\mathbf w} \leq 0$ and \begin{align*} \frac{w_{p^{j-1}}}{w_{p^j}} \leq p^{-\delta_{\mathbf w}}~\text{for all integers} ~~ j \geq 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 and satisfies the above inequality. The proof of the main result is a refinement of the techniques of Stetler.