{
  "id": "2408.16994",
  "version": "v1",
  "published": "2024-08-30T03:53:54.000Z",
  "updated": "2024-08-30T03:53:54.000Z",
  "title": "Nayak's theorem for compact operators",
  "authors": [
    "B V Rajarama Bhat",
    "Neeru Bala"
  ],
  "comment": "14 Pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $A$ be an $m\\times m$ complex matrix and let $\\lambda _1, \\lambda _2, \\ldots , \\lambda _m$ be the eigenvalues of $A$ arranged such that $|\\lambda _1|\\geq |\\lambda _2|\\geq \\cdots \\geq |\\lambda _m|$ and for $n\\geq 1,$ let $s^{(n)}_1\\geq s^{(n)}_2\\geq \\cdots \\geq s^{(n)}_m$ be the singular values of $A^n$. Then a famous theorem of Yamamoto (1967) states that $$\\lim _{n\\to \\infty}(s^{(n)}_j )^{\\frac{1}{n}}= |\\lambda _j|, ~~\\forall \\,1\\leq j\\leq m.$$ Recently S. Nayak strengthened this result very significantly by showing that the sequence of matrices $|A^n|^{\\frac{1}{n}}$ itself converges to a positive matrix $B$ whose eigenvalues are $|\\lambda _1|,|\\lambda _2|,$ $\\ldots , |\\lambda _m|.$ Here this theorem has been extended to arbitrary compact operators on infinite dimensional complex separable Hilbert spaces. The proof makes use of Nayak's theorem, Stone-Weirstrass theorem, Borel-Caratheodory theorem and some technical results of Anselone and Palmer on collectively compact operators. Simple examples show that the result does not hold for general bounded operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-30T03:53:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A10",
      "47B06",
      "47B07"
    ],
    "keywords": [
      "nayaks theorem",
      "dimensional complex separable hilbert spaces",
      "infinite dimensional complex separable hilbert",
      "arbitrary compact operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}