{
  "id": "1701.08826",
  "version": "v1",
  "published": "2017-01-30T21:05:30.000Z",
  "updated": "2017-01-30T21:05:30.000Z",
  "title": "Specht's criterion for systems of linear mappings",
  "authors": [
    "Vyacheslav Futorny",
    "Roger A. Horn",
    "Vladimir V. Sergeichuk"
  ],
  "comment": "21 pages",
  "journal": "Linear Algebra Appl. 519C (2017) 278-295",
  "doi": "10.1016/j.laa.2017.01.006",
  "categories": [
    "math.RT"
  ],
  "abstract": "W.Specht (1940) proved that two $n\\times n$ complex matrices $A$ and $B$ are unitarily similar if and only if $\\operatorname{trace} w(A,A^{\\ast}) = \\operatorname{trace} w(B,B^{\\ast})$ for every word $w(x,y)$ in two noncommuting variables. We extend his criterion and its generalizations by N.A.Wiegmann (1961) and N.Jing (2015) to an arbitrary system $\\mathcal A$ consisting of complex or real inner product spaces and linear mappings among them. We represent such a system by the directed graph $Q(\\mathcal A)$, whose vertices are inner product spaces and arrows are linear mappings. Denote by $\\widetilde Q(\\mathcal A)$ the directed graph obtained by enlarging to $Q(\\mathcal A)$ the adjoint linear mappings. We prove that a system $\\mathcal A$ is transformed by isometries of its spaces to a system $\\mathcal B$ if and only if the traces of all closed directed walks in $\\widetilde Q(\\mathcal A)$ and $\\widetilde Q(\\mathcal B)$ coincide.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-30T21:05:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A21",
      "15A63",
      "16G20",
      "47A67"
    ],
    "keywords": [
      "spechts criterion",
      "real inner product spaces",
      "directed graph",
      "adjoint linear mappings",
      "arbitrary system"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}