{
  "id": "2008.10058",
  "version": "v1",
  "published": "2020-08-23T15:42:10.000Z",
  "updated": "2020-08-23T15:42:10.000Z",
  "title": "On the spectral properties of the Hilbert transform operator on multi-intervals",
  "authors": [
    "Marco Bertola",
    "Alexander Katsevich",
    "Alexander Tovbis"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $J,E\\subset\\mathbb R$ be two multi-intervals with non-intersecting interiors. Consider the following operator $$A:\\, L^2( J )\\to L^2(E),\\ (Af)(x) = \\frac 1\\pi\\int_{ J } \\frac {f(y)\\text{d} y}{x-y},$$ and let $A^\\dagger$ be its adjoint. We introduce a self-adjoint operator $\\mathscr K$ acting on $L^2(E)\\oplus L^2(J)$, whose off-diagonal blocks consist of $A$ and $A^\\dagger$. In this paper we study the spectral properties of $\\mathscr K$ and the operators $A^\\dagger A$ and $A A^\\dagger$. Our main tool is to obtain the resolvent of $\\mathscr K$, which is denoted by $\\mathscr R$, using an appropriate Riemann-Hilbert problem, and then compute the jump and poles of $\\mathscr R$ in the spectral parameter $\\lambda$. We show that the spectrum of $\\mathscr K$ has an absolutely continuous component $[0,1]$ if and only if $J$ and $E$ have common endpoints, and its multiplicity equals to their number. If there are no common endpoints, the spectrum of $\\mathscr K$ consists only of eigenvalues and $0$. If there are common endpoints, then $\\mathscr K$ may have eigenvalues imbedded in the continuous spectrum, each of them has a finite multiplicity, and the eigenvalues may accumulate only at $0$. In all cases, $\\mathscr K$ does not have a singular continuous spectrum. The spectral properties of $A^\\dagger A$ and $A A^\\dagger$, which are very similar to those of $\\mathscr K$, are obtained as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-23T15:42:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral properties",
      "hilbert transform operator",
      "common endpoints",
      "multi-intervals",
      "appropriate riemann-hilbert problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}