{
  "id": "math/0612487",
  "version": "v1",
  "published": "2006-12-18T17:27:53.000Z",
  "updated": "2006-12-18T17:27:53.000Z",
  "title": "Generalized Krein algebras and asymptotics of Toeplitz determinants",
  "authors": [
    "Albrecht Böttcher",
    "Alexei Karlovich",
    "Bernd Silbermann"
  ],
  "comment": "27 pages",
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "We give a survey on generalized Krein algebras $K_{p,q}^{\\alpha,\\beta}$ and their applications to Toeplitz determinants. Our methods originated in a paper by Mark Krein of 1966, where he showed that $K_{2,2}^{1/2,1/2}$ is a Banach algebra. Subsequently, Widom proved the strong Szeg\\H{o} limit theorem for block Toeplitz determinants with symbols in $(K_{2,2}^{1/2,1/2})_{N\\times N}$ and later two of the authors studied symbols in the generalized Krein algebras $(K_{p,q}^{\\alpha,\\beta})_{N\\times N}$, where $\\lambda:=1/p+1/q=\\alpha+\\beta$ and $\\lambda=1$. We here extend these results to $0<\\lambda<1$. The entire paper is based on fundamental work by Mark Krein, ranging from operator ideals through Toeplitz operators up to Wiener-Hopf factorization.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-12-18T17:27:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35",
      "15A15",
      "47B10"
    ],
    "keywords": [
      "generalized krein algebras",
      "mark krein",
      "asymptotics",
      "block toeplitz determinants",
      "limit theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....12487B"
    }
  }
}