{
  "id": "2103.09698",
  "version": "v1",
  "published": "2021-03-17T14:49:01.000Z",
  "updated": "2021-03-17T14:49:01.000Z",
  "title": "On the orthogonality of generalized eigenspaces for the Ornstein--Uhlenbeck operator",
  "authors": [
    "Valentina Casarino",
    "Paolo Ciatti",
    "Peter Sjögren"
  ],
  "comment": "10 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We study the orthogonality of the generalized eigenspaces of an Ornstein--Uhlenbeck operator $\\mathcal L$ in $\\mathbb{R}^N$, with drift given by a real matrix $B$ whose eigenvalues have negative real parts. If $B$ has only one eigenvalue, we prove that any two distinct generalized eigenspaces of $\\mathcal L$ are orthogonal with respect to the invariant Gaussian measure. Then we show by means of two examples that if $B$ admits distinct eigenvalues, the generalized eigenspaces of $\\mathcal L$ may or may not be orthogonal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-17T14:49:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A18",
      "47A70",
      "47D03"
    ],
    "keywords": [
      "ornstein-uhlenbeck operator",
      "orthogonality",
      "admits distinct eigenvalues",
      "invariant gaussian measure",
      "real matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}