{
  "id": "2208.07569",
  "version": "v1",
  "published": "2022-08-16T07:25:06.000Z",
  "updated": "2022-08-16T07:25:06.000Z",
  "title": "Bounded analytic functions on certain symmetrized domains",
  "authors": [
    "Mainak Bhowmik",
    "Poornendu Kumar"
  ],
  "comment": "20 page; Comments are welcome",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this note, we first give a canonical structure of scalar-valued rational inner functions on the symmetrized polydisc. Then we generalize Caratheodory's approximation result to the setting of matrix-valued holomorphic functions on the symmetrized bidisc, $\\mathbb{G}$. We approximate matrix-valued holomorphic functions with sup-norm not greater than one by rational iso-inner or coiso-inner functions uniformly on compact subsets of the symmetrized bidisc using the interpolation results. En route, we also prove that any solvable data with initial nodes on $\\mathbb{G}$ and the final nodes in the operator norm unit ball of the rectangular matrices has a rational iso-inner or coiso-inner solution. Finally, we give necessary and sufficient conditions for matrix-valued Schur class functions on $\\mathbb{G}$ to have a Schur class factorization on $\\mathbb{G}$ in several situations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-16T07:25:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A48",
      "47A68",
      "47A56",
      "32A17",
      "32E30"
    ],
    "keywords": [
      "bounded analytic functions",
      "symmetrized domains",
      "rational iso-inner",
      "operator norm unit ball",
      "schur class factorization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}