{
  "id": "2503.07069",
  "version": "v2",
  "published": "2025-03-10T08:52:54.000Z",
  "updated": "2025-06-20T07:43:22.000Z",
  "title": "The isoperimetric inequality for partial sums of Toeplitz eigenvalues in the Fock space",
  "authors": [
    "Fabio Nicola",
    "Federico Riccardi",
    "Paolo Tilli"
  ],
  "comment": "25 pages. Title changed. The presentation of the arguments has been improved. Added a part concerning a \"breaking symmetry\" phenomenon for the second eigenvalue (see Proposition 1.4)",
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "We prove that, among all subsets $\\Omega\\subset \\mathbb{C}$ having circular symmetry and prescribed measure, the ball is the only maximizer of the sum of the first $K$ eigenvalues ($K\\geq 1$) of the corresponding Toeplitz operator $T_\\Omega$ on the Fock space $\\mathcal{F}$. As a byproduct, we prove that balls maximize any Schatten $p$-norm of $T_\\Omega$ for $p>1$ (and minimize the corresponding quasinorm for $p<1$), and that the second eigenvalue is maximized by a particular annulus. Moreover, we extend some of these results to general radial symbols in $L^p(\\mathbb{C})$, with $p > 1$, characterizing those that maximize the sum of the first $K$ eigenvalues. We also show a symmetry breaking phenomenon for the second eigenvalue, when the assumption of circular symmetry is dropped.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-06-20T07:43:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fock space",
      "partial sums",
      "toeplitz eigenvalues",
      "isoperimetric inequality",
      "circular symmetry"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}