{
  "id": "1405.0860",
  "version": "v2",
  "published": "2014-05-05T11:21:26.000Z",
  "updated": "2014-09-08T09:50:41.000Z",
  "title": "On Borel equivalence relations related to self-adjoint operators",
  "authors": [
    "Hiroshi Ando",
    "Yasumichi Matsuzawa"
  ],
  "comment": "10 pages, added more detail of the proof of Proposition 3.8 after the referee's suggestion",
  "categories": [
    "math.LO",
    "math.FA",
    "math.SP"
  ],
  "abstract": "In a recent work, the authors studied various Borel equivalence relations defined on the Polish space ${\\rm{SA}}(H)$ of all (not necessarily bounded) self-adjoint operators on a separable infinite-dimensional Hilbert space $H$. In this paper we study the domain equivalence relation $E_{\\rm{dom}}^{{\\rm{SA}}(H)}$ given by $AE_{\\rm{dom}}^{{\\rm{SA}}(H)}B\\Leftrightarrow {\\rm{dom}}{A}={\\rm{dom}}{B}$ and determine its exact Borel complexity: $E_{\\rm{dom}}^{{\\rm{SA}}(H)}$ is an $F_{\\sigma}$ (but not $K_{\\sigma}$) equivalence relation which is continuously bireducible with the orbit equivalence relation $E_{\\ell^{\\infty}}^{\\mathbb{R}^{\\mathbb{N}}}$ of the standard Borel group $\\ell^{\\infty}=\\ell^{\\infty}(\\mathbb{N},\\mathbb{R})$ on $\\mathbb{R}^{\\mathbb{N}}$. This, by Rosendal's Theorem, shows that $E_{\\rm{dom}}^{{\\rm{SA}}(H)}$ is universal for $K_{\\sigma}$ equivalence relations. Moreover, we show that generic self-adjoint operators have purely singular continuous spectrum equal to $\\mathbb{R}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-05T11:21:26.000Z",
      "comment": "8 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-08T09:50:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "borel equivalence relations",
      "purely singular continuous spectrum equal",
      "domain equivalence relation",
      "separable infinite-dimensional hilbert space",
      "generic self-adjoint operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.0860A"
    }
  }
}