{
  "id": "1901.09254",
  "version": "v1",
  "published": "2019-01-26T17:56:44.000Z",
  "updated": "2019-01-26T17:56:44.000Z",
  "title": "Equivalence after extension and Schur coupling do not coincide, on essentially incomparable Banach spaces",
  "authors": [
    "Sanne ter Horst",
    "Miek Messerschmidt",
    "Andre C. M. Ran",
    "Mark Roelands"
  ],
  "comment": "10 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In 1994 H. Bart and V.\\'{E}. Tsekanovskii posed the question whether the Banach space operator relations matricial coupling (MC), equivalence after extension (EAE) and Schur coupling (SC) coincide, leaving only the implication EAE/MC $\\Rightarrow$ SC open. Despite several affirmative results, in this paper we show that the answer in general is no. This follows from a complete description of EAE and SC for the case that the operators act on essentially incomparable Banach spaces, which also leads to a new characterization of the notion of essential incomparability. Concretely, the forward shift operators $U$ on $\\ell^p$ and $V$ on $\\ell^q$, for $1\\leq p,q\\leq \\infty$, $p\\neq q$, are EAE but not SC. As a corollary, SC is not transitive. Under mild assumptions, given $U$ and $V$ that are Atkinson or generalized invertible and EAE, we give a concrete operator $W$ that is SC to both $U$ and $V$, even if $U$ and $V$ are not SC themselves. Some further affirmative results for the case where the Banach spaces are isomorphic are also obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-26T17:56:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A62",
      "47A53"
    ],
    "keywords": [
      "essentially incomparable banach spaces",
      "schur coupling",
      "operator relations matricial coupling",
      "banach space operator relations matricial",
      "equivalence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}