{
  "id": "2010.06585",
  "version": "v1",
  "published": "2020-10-13T11:18:36.000Z",
  "updated": "2020-10-13T11:18:36.000Z",
  "title": "Non-commutative rational functions in the full Fock space",
  "authors": [
    "Michael T. Jury",
    "Robert T. W. Martin",
    "Eli Shamovich"
  ],
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "A rational function belongs to the Hardy space, $H^2$, of square-summable power series if and only if it is bounded in the complex unit disk. Any such rational function is necessarily analytic in a disk of radius greater than one. The inner-outer factorization of a rational function, $\\mathfrak{r} \\in H^2$ is particularly simple: The inner factor of $\\mathfrak{r}$ is a (finite) Blaschke product and (hence) both the inner and outer factors are again rational. We extend these and other basic facts on rational functions in $H^2$ to the full Fock space over $\\mathbb{C}^d$, identified as the \\emph{non-commutative (NC) Hardy space} of square-summable power series in several NC variables. In particular, we characterize when an NC rational function belongs to the Fock space, we prove analogues of classical results for inner-outer factorizations of NC rational functions and NC polynomials, and we obtain spectral results for NC rational multipliers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-13T11:18:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "full fock space",
      "non-commutative rational functions",
      "square-summable power series",
      "inner-outer factorization",
      "nc rational function belongs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}