{
  "id": "1207.1947",
  "version": "v1",
  "published": "2012-07-09T04:48:48.000Z",
  "updated": "2012-07-09T04:48:48.000Z",
  "title": "Compactness and an approximation property related to an operator ideal",
  "authors": [
    "Anil Kumar Karn",
    "Deba Prasad Sinha"
  ],
  "comment": "23 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "For an operator ideal $\\mathcal A$, we study the composition operator ideals ${\\mathcal A}\\circ{\\mathcal K}$, ${\\mathcal K}\\circ{\\mathcal A}$ and ${\\mathcal K}\\circ{\\mathcal A}\\circ{\\mathcal K}$, where $\\mathcal K$ is the ideal of compact operators. We introduce a notion of an $\\mathcal A$-approximation property on a Banach space and characterise it in terms of the density of finite rank operators in ${\\mathcal A}\\circ{\\mathcal K}$ and ${\\mathcal K}\\circ{\\mathcal A}$. We propose the notions of $\\ell_{\\infty}$-extension and $\\ell_{1}$-lifting properties for an operator ideal $\\mathcal A$ and study ${\\mathcal A}\\circ{\\mathcal K}$, ${\\mathcal}\\circ{\\mathcal A}$ and the $\\mathcal A$-approximation property where $\\mathcal A$ is injective or surjective and/or with the $\\ell_{\\infty}$-extension or $\\ell_{1}$-lifting property. In particular, we show that if $\\mathcal A$ is an injective operator ideal with the $\\ell_\\infty$-extension property, then we have: (a) $X$ has the $\\mathcal A$-approximation property if and only if $({\\mathcal A}^{min})^{inj}(Y,X)={\\mathcal A}^{min}(Y,X)$, for all Banach spaces $Y$. (b) The dual space $X^*$ has the $\\mathcal A$-approximation property if and only if $(({\\mathcal A}^{dual})^{min})^{sur}(X,Y)=({\\mathcal A}^{dual})^{min}(X,Y)$, for all Banach spaces $Y$.}For an operator ideal $\\mathcal A$, we study the composition operator ideals ${\\mathcal A}\\circ{\\mathcal K}$,",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-07-09T04:48:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B50",
      "46B20",
      "46B28",
      "47B07"
    ],
    "keywords": [
      "approximation property",
      "composition operator ideals",
      "banach space",
      "compactness",
      "finite rank operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.1947K"
    }
  }
}