{
  "id": "2306.16141",
  "version": "v1",
  "published": "2023-06-28T12:16:02.000Z",
  "updated": "2023-06-28T12:16:02.000Z",
  "title": "On the convexity of spatial numetical range in normed algebras",
  "authors": [
    "H. V. Dedania",
    "A. B. Patel"
  ],
  "comment": "9 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this article, we address the following question: Is it true that the spatial numerical range (SNR) $V_A(a)$ of an element $a$ in a normed algebra $(A, \\|\\cdot\\|)$ is always convex? If the normed algebra is unital, then it is convex \\cite[Theorem 3, P.16]{BoDu:71}. In non-unital case, we believe that the problem is still open and its answer seems to be negative. In search of such a normed algebra, we have proved that the SNR $V_A(a)$ is convex in several non-unital Banach algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-28T12:16:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46H05",
      "47A12"
    ],
    "keywords": [
      "normed algebra",
      "spatial numetical range",
      "non-unital banach algebras",
      "spatial numerical range",
      "non-unital case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}