{
  "id": "2108.13710",
  "version": "v1",
  "published": "2021-08-31T09:48:41.000Z",
  "updated": "2021-08-31T09:48:41.000Z",
  "title": "Cross-Toeplitz Operators on the Fock--Segal--Bargmann Spaces and Two-Sided Convolutions on the Heisenberg Group",
  "authors": [
    "Vladimir V. Kisil"
  ],
  "comment": "44 p., AMS-LateX, 3 PDF images in two figures",
  "categories": [
    "math.FA",
    "math.CV",
    "math.OA",
    "math.RT",
    "quant-ph"
  ],
  "abstract": "We introduce an extended class of cross-Toeplitz operators which act between Fock--Segal--Bargmann spaces with different weights. It is natural to consider these operators in the framework of representation theory of the Heisenberg group. Our main technique is representation of cross-Toeplitz by two-sided relative convolutions from the Heisenberg group. In turn, two-sided convolutions are reduced to usual (one-sided) convolutions on the Heisenberg group of the doubled dimensionality. This allows us to utilise the powerful group-representation technique of coherent states, co- and contra-variant transforms, twisted convolutions, symplectic Fourier transform, etc.We discuss connections of (cross-)Toeplitz operators with pseudo-differential operators, localisation operators in time-frequency analysis, and characterisation of kernels in terms of ladder operators. The paper is written in detailed and reasonably self-contained manner to be suitable as an introduction into group-theoretical methods in phase space and time-frequency operator theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-31T09:48:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35",
      "30H20",
      "43A15",
      "44A35",
      "46E22",
      "47B32",
      "47G30",
      "81R30",
      "81S30"
    ],
    "keywords": [
      "heisenberg group",
      "cross-toeplitz operators",
      "fock-segal-bargmann spaces",
      "two-sided convolutions",
      "time-frequency operator theory"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}