{
  "id": "math/0011232",
  "version": "v1",
  "published": "2000-11-28T09:29:01.000Z",
  "updated": "2000-11-28T09:29:01.000Z",
  "title": "Coordinate restrictions of linear operators in $l_2^n$",
  "authors": [
    "R. Vershynin"
  ],
  "categories": [
    "math.FA",
    "math.AP",
    "math.PR"
  ],
  "abstract": "This paper addresses the problem of improving properties of a linear operator u in $l_2^n$ by restricting it onto coordinate subspaces. We discuss how to reduce the norm of u by a random coordinate restriction, how to approximate u by a random operator with small \"coordinate\" rank, how to find coordinate subspaces where u is an isomorphism. The first problem in this list provides a probabilistic extension of a suppression theorem of Kashin and Tzafriri, the second one is a new look at a result of Rudelson on the random vectors in the isotropic position, the last one is the recent generalization of the Bourgain-Tzafriri's invertibility principle. The main point is that all the results are independent of n, the situation is instead controlled by the Hilbert-Schmidt norm of u. As an application, we provide an almost optimal solution to the problem of harmonic density in harmonic analysis, and a solution to the reconstruction problem for communication networks which deliver data with random losses.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-11-28T09:29:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B09",
      "60G50",
      "43A46",
      "43A46"
    ],
    "keywords": [
      "linear operator",
      "coordinate subspaces",
      "random coordinate restriction",
      "bourgain-tzafriris invertibility principle",
      "harmonic analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math.....11232V"
    }
  }
}