{
  "id": "2405.19964",
  "version": "v1",
  "published": "2024-05-30T11:40:59.000Z",
  "updated": "2024-05-30T11:40:59.000Z",
  "title": "A Proof of $\\mathfrak{L}^2$-Boundedness for Magnetic Pseudodifferential Super Operators via Matrix Representations With Respect to Parseval Frames",
  "authors": [
    "Gihyun Lee",
    "Max Lein"
  ],
  "comment": "35 pages",
  "categories": [
    "math-ph",
    "math.FA",
    "math.MP"
  ],
  "abstract": "A fundamental result in pseudodifferential theory is the Calder\\'on-Vaillancourt theorem, which states that a pseudodifferential operator defined from a H\\\"ormander symbol of order $0$ defines a bounded operator on $L^2(\\mathbb{R}^d)$. In this work we prove an analog for pseudodifferential \\emph{super} operator, \\ie operators acting on other operators, in the presence of magnetic fields. More precisely, we show that magnetic pseudodifferential super operators of order $0$ define bounded operators on the space of Hilbert-Schmidt operators $\\mathfrak{L}^2 \\bigl ( \\mathcal{B} \\bigl ( L^2(\\mathbb{R}^d) \\bigr ) \\bigr )$. Our proof is inspired by the recent work of Cornean, Helffer and Purice and rests on a characterization of magnetic pseudodifferential super operators in terms of their \"matrix element\" computed with respect to a Parseval frame.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-30T11:40:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35S05",
      "46B15",
      "47C15",
      "47L80"
    ],
    "keywords": [
      "magnetic pseudodifferential super operators",
      "parseval frame",
      "matrix representations",
      "boundedness",
      "magnetic fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}