Multipliers of the Hilbert spaces of Dirichlet series

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 ${\mathcal{H}}_{\mathbf{w}}$ denote the reproducing kernel Hilbert space associated with ${\kappa }_{\mathbf{w}}.$ Let \begin{equation*} δ_{\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 $\{p_j{\}}_{j⩾1}$ is an increasing enumeration of prime numbers and $\text{gpf}(n)$ denotes the greatest prime factor of an integer $n⩾2.$ If $\mathbf{w}$ satisfies \begin{equation*} \sum_{\substack{j \geqslant 2\\ j | n}} j^{-δ_\mathbf w} w_j μ\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 $\mathcal{M}({\mathcal{H}}_{\mathbf{w}})$ of ${\mathcal{H}}_{\mathbf{w}}$ 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 $\mathcal{M}({\mathcal{H}}_{\mathbf{w}})$ when $\mathbf{w}$ is an additive function satisfying ${\delta }_{\mathbf{w}}⩽0$ and \begin{align*} \frac{w_{p^{j-1}}}{w_{p^j}} \leqslant p^{-δ_{\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 ${\mathcal{H}}_{\mathbf{w}}$ when $\mathbf{w}$ is multiplicative. The proof of the main result is a refinement of the techniques of Stetler.

arXiv commentMinor changes and corrections. To appear in NYJM