{
  "id": "1409.3497",
  "version": "v1",
  "published": "2014-09-11T16:39:05.000Z",
  "updated": "2014-09-11T16:39:05.000Z",
  "title": "Metric operators, generalized hermiticity and lattices of Hilbert lpaces",
  "authors": [
    "Jean-Pierre Antoine",
    "Camillo Trapani"
  ],
  "comment": "51pages; will appear as a chapter in \\textit{Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects}; F. Bagarello, J-P. Gazeau, F. H. Szafraniec and M. Znojil, eds., J. Wiley, 2015",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (PIP-space). We introduce several generalizations of the notion of similarity between operators, in particular, the notion of quasi-similarity, and we explore to what extend they preserve spectral properties. Then we apply some of the previous results to operators on a particular PIP-space, namely, a scale of Hilbert spaces generated by a metric operator. Finally, motivated by the recent developments of pseudo-Hermitian quantum mechanics, we reformulate the notion of pseudo-Hermitian operators in the preceding formalism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-11T16:39:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hilbert lpaces",
      "generalized hermiticity",
      "hilbert space",
      "partial inner product space",
      "pseudo-hermitian quantum mechanics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.3497A"
    }
  }
}