{
  "id": "1408.0525",
  "version": "v1",
  "published": "2014-08-03T18:40:01.000Z",
  "updated": "2014-08-03T18:40:01.000Z",
  "title": "A pre-order on positive real operators and its invariance under linear fractional transformations",
  "authors": [
    "S. ter Horst"
  ],
  "comment": "15 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "A pre-order and equivalence relation on the class of positive real Hilbert space operators are introduced, in correspondence with similar relations for contraction operators defined by Yu.L. Shmul'yan in [7]. It is shown that the pre-order, and hence the equivalence relation, are preserved by certain linear fractional transformations. As an application, the operator relations are extended to the class $\\fC(\\cU)$ of Carath\\'eodory functions on the unit disc $\\BD$ of $\\BC$ whose values are operators on a finite dimensional Hilbert space $\\cU$. With respect to these relations on $\\fC(\\cU)$ it turns out that the associated linear fractional transformations of $\\fC(\\cU)$ preserve the equivalence relation on their natural domain of definition, but not necessarily the pre-order, paralleling similar results for Schur class functions in [3].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-03T18:40:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A62",
      "47A56",
      "47A57"
    ],
    "keywords": [
      "linear fractional transformations",
      "positive real operators",
      "equivalence relation",
      "finite dimensional hilbert space",
      "invariance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.0525T"
    }
  }
}