{
  "id": "2202.01745",
  "version": "v1",
  "published": "2022-02-03T18:09:47.000Z",
  "updated": "2022-02-03T18:09:47.000Z",
  "title": "Symmetric matrix representations of truncated Toeplitz operators on finite dimensional spaces",
  "authors": [
    "Ryan O'Loughlin"
  ],
  "categories": [
    "math.FA",
    "math.AG"
  ],
  "abstract": "In this paper, we study matrix representations of truncated Toeplitz operators with respect to orthonormal bases which are invariant under a canonical conjugation map. In particular, we determine necessary and sufficient conditions for when a 3-by-3 symmetric matrix is the matrix representation of a truncated Toeplitz operator with respect to a given conjugation invariant orthonormal basis. We specialise our result to the case when the conjugation invariant orthonormal basis is a modified Clark basis. As a corollary to this specialisation, we answer a previously stated open conjecture in the negative, and show that not every unitary equivalence between a complex symmetric matrix and a truncated Toeplitz operator arises from a modified Clark basis representation. Finally, we show that a given 3-by-3 symmetric matrix is the matrix representation of a truncated Toeplitz operator with respect to a conjugation invariant orthonormal basis if and only if a specified system of polynomial equations is satisfied with a real solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-03T18:09:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30H10",
      "47B35",
      "46E20"
    ],
    "keywords": [
      "truncated toeplitz operator",
      "conjugation invariant orthonormal basis",
      "symmetric matrix representations",
      "finite dimensional spaces",
      "modified clark basis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}