{
  "id": "2411.13804",
  "version": "v1",
  "published": "2024-11-21T03:08:30.000Z",
  "updated": "2024-11-21T03:08:30.000Z",
  "title": "The Brown Measure of Non-Hermitian Sums of Projections",
  "authors": [
    "Max Sun Zhou"
  ],
  "comment": "31 pages, 2 figures",
  "categories": [
    "math.OA",
    "math.PR"
  ],
  "abstract": "We compute the Brown measure of the non-normal operators $X = p + i q$, where $p$ and $q$ are Hermitian, freely independent, and have spectra consisting of $2$ atoms. The computation relies on the model of the non-trivial part of the von Neumann algebra generated by 2 projections as $2 \\times 2$ random matrices. We observe that these measures are supported on hyperbolas and note some other properties related to their atoms and symmetries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-21T03:08:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "brown measure",
      "non-hermitian sums",
      "projections",
      "von neumann algebra",
      "computation relies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}