{
  "id": "2212.08182",
  "version": "v1",
  "published": "2022-12-15T22:48:41.000Z",
  "updated": "2022-12-15T22:48:41.000Z",
  "title": "Diagonals of self-adjoint operators",
  "authors": [
    "Marcin Bownik",
    "John Jasper"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Given a self-adjoint operator $T$ on a separable infinite-dimensional Hilbert space we study the problem of characterizing the set $\\mathcal D(T)$ of all possible diagonals of $T$. For compact operators $T$, we give a complete characterization of diagonals modulo the kernel of $T$. That is, we characterize $\\mathcal D(T)$ for the class of operators sharing the same nonzero eigenvalues (with multiplicities) as $T$. Moreover, we determine $\\mathcal D(T)$ for a fixed compact operator $T$, modulo the kernel problem for positive compact operators with finite-dimensional kernel. For operators $T$ with at least two points in their essential spectrum $\\sigma_{ess}(T)$, we give a complete characterization of $\\mathcal D(T)$ for the class of self-adjoint operators sharing the same spectral measure as $T$ with a possible exception of multiplicities of eigenvalues at the extreme points of $\\sigma_{ess}(T)$. As in the case of compact operators, we also give a more precise description of $\\mathcal D(T)$ for a fixed self-adjoint operator $T$, albeit modulo the kernel problem for special classes of operators. These classes consist of operators $T$ for which an extreme point of the essential spectrum $\\sigma_{ess}(T)$ is also an extreme point of the spectrum $\\sigma(T)$. Our results generalize a characterization of diagonals of compact positive operators by Kaftal, Loreaux, and Weiss, Blaschke-type results of M\\\"uller and Tomilov, and a characterization of diagonals of operators with finite spectrum by the authors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-15T22:48:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "self-adjoint operator",
      "extreme point",
      "essential spectrum",
      "complete characterization",
      "kernel problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}