{
  "id": "2411.07324",
  "version": "v1",
  "published": "2024-11-11T19:33:17.000Z",
  "updated": "2024-11-11T19:33:17.000Z",
  "title": "The Hilbert matrix done right",
  "authors": [
    "A. Montes-Rodríguez",
    "J. A. Virtanen"
  ],
  "comment": "To appear in Operator Theory: Advances and Applications (IWOTA 2023)",
  "categories": [
    "math.FA",
    "math.CV"
  ],
  "abstract": "We give very simple proofs of the classical results of Magnus and Hill on the spectral properties of the Hilbert matrix $$ H = \\left ( {1 \\over i+j+ 1 } \\right )_{i,j\\geq 0} $$ which defines a bounded linear operator on the sequence space $\\ell^2$. In particular, we use the Mehler-Fock transform to find the spectrum and the latent eigenfunctions of the Hilbert matrix, that is, we show that the spectrum of $H$ is $[0,\\pi]$ with no eigenvalues (Magnus' result) and describe all complex sequences $x$ such that $Hx=\\mu x$ for some complex number $\\mu$ (Hill's result).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-11T19:33:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hilbert matrix",
      "simple proofs",
      "complex sequences",
      "hills result",
      "spectral properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}