{
  "id": "1708.08633",
  "version": "v1",
  "published": "2017-08-29T08:14:46.000Z",
  "updated": "2017-08-29T08:14:46.000Z",
  "title": "Remarks on the Crouzeix-Palencia proof that the numerical range is a $(1+\\sqrt2)$-spectral set",
  "authors": [
    "Thomas Ransford",
    "Felix Schwenninger"
  ],
  "comment": "4 pages",
  "categories": [
    "math.FA",
    "math.NA"
  ],
  "abstract": "Crouzeix and Palencia recently showed that the numerical range of a Hilbert-space operator is a $(1+\\sqrt2)$-spectral set for the operator. One of the principal ingredients of their proof can be formulated as an abstract functional-analysis lemma. We give a new short proof of the lemma and show that, in the context of this lemma, the constant $(1+\\sqrt2)$ is sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-29T08:14:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A25",
      "47A12"
    ],
    "keywords": [
      "spectral set",
      "numerical range",
      "crouzeix-palencia proof",
      "abstract functional-analysis lemma",
      "hilbert-space operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}