{
  "id": "2301.06413",
  "version": "v1",
  "published": "2023-01-16T13:06:28.000Z",
  "updated": "2023-01-16T13:06:28.000Z",
  "title": "Multipliers of the Hilbert spaces of Dirichlet series",
  "authors": [
    "Chaman Kumar Sahu"
  ],
  "categories": [
    "math.FA"
  ],
  "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 \\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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-16T13:06:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30B50",
      "46E22",
      "11Z05"
    ],
    "keywords": [
      "real number",
      "multiplier algebra",
      "reproducing kernel hilbert space",
      "greatest prime factor",
      "right half plane"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}