{
  "id": "2402.05503",
  "version": "v1",
  "published": "2024-02-08T09:21:29.000Z",
  "updated": "2024-02-08T09:21:29.000Z",
  "title": "On the type of ill-posedness of generalized Hilbert matrices and related operators",
  "authors": [
    "Stefan Kindermann"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We consider infinite-dimensional generalized Hilbert matrices of the form $H_{i,j} = \\frac{d_i d_j}{x_i + x_j}$, where $d_i$ are nonnegative weights and $x_i$ are pairwise disjoint positive numbers. We state sufficient and, for monotonically rearrangeable $x_i$, also necessary conditions for $d_i$, $x_i$ such that the induced operator from $\\ell^2 \\to \\ell^2$ and related operators are well-defined, bounded, or compact. Furthermore, we give conditions, when this operator is injective and ill-posed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-08T09:21:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "related operators",
      "ill-posedness",
      "infinite-dimensional generalized hilbert matrices",
      "pairwise disjoint positive numbers",
      "state sufficient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}