{
  "id": "0809.3047",
  "version": "v1",
  "published": "2008-09-18T01:46:19.000Z",
  "updated": "2008-09-18T01:46:19.000Z",
  "title": "Commutators on $\\ell_1$",
  "authors": [
    "Detelin Dosev"
  ],
  "comment": "17 pages. Submitted to the Journal of Functional Analysis",
  "doi": "10.1112/blms/bdp110",
  "categories": [
    "math.FA"
  ],
  "abstract": "The main result is that the commutators on $\\ell_1$ are the operators not of the form $\\lambda I + K$ with $\\lambda\\neq 0$ and $K$ compact. We generalize Apostol's technique (1972, Rev. Roum. Math. Appl. 17, 1513 - 1534) to obtain this result and use this generalization to obtain partial results about the commutators on spaces $\\X$ which can be represented as $\\displaystyle \\X\\simeq (\\bigoplus_{i=0}^{\\infty} \\X)_{p}$ for some $1\\leq p<\\infty$ or $p=0$. In particular, it is shown that every compact operator on $L_1$ is a commutator. A characterization of the commutators on $\\ell_{p_1}\\oplus\\ell_{p_2}\\oplus...\\oplus\\ell_{p_n}$ is given. We also show that strictly singular operators on $\\ell_{\\infty}$ are commutators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-09-18T01:46:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B47"
    ],
    "keywords": [
      "commutator",
      "main result",
      "partial results",
      "compact operator",
      "generalize apostols technique"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.3047D"
    }
  }
}