{
  "id": "1902.09869",
  "version": "v1",
  "published": "2019-02-26T11:24:40.000Z",
  "updated": "2019-02-26T11:24:40.000Z",
  "title": "Hilbert-Schmidt and Trace Class Pseudo-differential Operators on the Abstract Heisenberg Group",
  "authors": [
    "Aparajita Dasgupta",
    "Vishvesh Kumar"
  ],
  "comment": "16 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper we introduce and study pseudo-differential operators with operator valued symbols on the abstract Heisenberg group $\\mathbb{H}(G):=G \\times \\widehat{G} \\times \\mathbb{T},$ where $G$ a locally compact abelian group with its dual group $\\widehat{G}$. We obtain a necessary and sufficient condition on symbols for which these operators are in the class of Hilbert-Schmidt operators. As a key step in proving this we derive a trace formula for the trace class $j$-Weyl transform, $j \\in \\mathbb{Z}^*$ with symbols in $L^{2}(G\\times \\widehat{G}).$ We go on to present a characterization of the trace class pseudo-differential operators on $\\mathbb{H}(G)$. Finally, we also give a trace formula for these trace class operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-26T11:24:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35S05",
      "47G30",
      "43A85",
      "43A77"
    ],
    "keywords": [
      "trace class pseudo-differential operators",
      "abstract heisenberg group",
      "hilbert-schmidt",
      "trace formula",
      "study pseudo-differential operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}