{
  "id": "2009.01300",
  "version": "v1",
  "published": "2020-09-02T18:55:20.000Z",
  "updated": "2020-09-02T18:55:20.000Z",
  "title": "Convexity of the orbit-closed $C$-numerical range and majorization",
  "authors": [
    "Jireh Loreaux",
    "Sasmita Patnaik"
  ],
  "comment": "44 pages, 2 figures",
  "categories": [
    "math.FA"
  ],
  "abstract": "We introduce and investigate the orbit-closed $C$-numerical range, a natural modification of the $C$-numerical range of an operator introduced for $C$ trace-class by Dirr and vom Ende. Our orbit-closed $C$-numerical range is a conservative modification of theirs because these two sets have the same closure and even coincide when $C$ is finite rank. Since Dirr and vom Ende's results concerning the $C$-numerical range depend only on its closure, our orbit-closed $C$-numerical range inherits these properties, but we also establish more. For $C$ selfadjoint, Dirr and vom Ende were only able to prove that the closure of their $C$-numerical range is convex, and asked whether it is convex without taking the closure. We establish the convexity of the orbit-closed $C$-numerical range for selfadjoint $C$ without taking the closure by providing a characterization in terms of majorization, unlocking the door to a plethora of results which generalize properties of the $C$-numerical range known in finite dimensions or when $C$ has finite rank. Under rather special hypotheses on the operators, we also show the $C$-numerical range is convex, thereby providing a partial answer to the question posed by Dirr and vom Ende.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-02T18:55:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12",
      "47B15",
      "52A10",
      "52A40",
      "26D15"
    ],
    "keywords": [
      "numerical range",
      "majorization",
      "finite rank",
      "finite dimensions",
      "special hypotheses"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}