{
  "id": "2408.04794",
  "version": "v1",
  "published": "2024-08-09T00:00:13.000Z",
  "updated": "2024-08-09T00:00:13.000Z",
  "title": "A regularity condition under which integral operators with operator-valued kernels are trace class",
  "authors": [
    "John Zweck",
    "Yuri Latushkin",
    "Erika Gallo"
  ],
  "comment": "27 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We study integral operators on the space of square-integrable functions from a compact set, $X$, to a separable Hilbert space, $H$. The kernel of such an operator takes values in the ideal of Hilbert-Schmidt operators on $H$. We establish regularity conditions on the kernel under which the associated integral operator is trace class. First, we extend Mercer's theorem to operator-valued kernels by proving that a continuous, nonnegative-definite, Hermitian symmetric kernel defines a trace class integral operator on $L^2(X;H)$ under an additional assumption. Second, we show that a general operator-valued kernel that is defined on a compact set and that is H\\\"older continuous with H\\\"older exponent greater than a half is trace class provided that the operator-valued kernel is essentially bounded as a mapping into the space of trace class operators on $H$. Finally, when $\\dim H < \\infty$, we show that an analogous result also holds for matrix-valued kernels on the real line, provided that an additional exponential decay assumption holds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-09T00:00:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "operator-valued kernel",
      "regularity condition",
      "additional exponential decay assumption holds",
      "trace class integral operator",
      "compact set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}