{
  "abstract": "Let $η_{1},\\cdots,η_{d}$ be nonconstant one-variable inner functions satisfying $η_{i}(0)=0$ for all $1\\leq i\\leq d$. Denote by $M_η$ the submodule of the Hardy space $H^{2}(\\mathbb{D}^{d})$ generated by the function family $\\{η_{i}(z_{i})-η_{i+1}(z_{i+1})\\mid 1\\leq i\\leq d-1\\}$. For the corresponding quotient module $N_η=H^{2}(\\mathbb{D}^{d})\\ominus M_η$, we define the compressed multiplication operators $S_{z_{i}}=P_{N_η}M_{z_{i}}\\vert_{N_η}$, where $P_{N_η}$ is the orthogonal projection from $H^{2}(\\mathbb{D}^{d})$ onto $N_η$. In this paper, we characterize the Hilbert-Schmidt property of the self-commutators $[S_{z_{i}}^{\\ast},S_{z_{i}}]$ and cross-commutators $[S_{z_{i}}^{\\ast},S_{z_{j}}]$ with $i\\neq j$. Moreover, explicit formulas for their Hilbert-Schmidt norms are established whenever these commutators are Hilbert-Schmidt.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02692",
  "author_total": 2,
  "authors": [
    {
      "name": "Anjian Xu"
    },
    {
      "name": "Yuchao Xiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bs485f9c5dnk5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02692",
  "fulltext_url": null,
  "id": "bs485f9c5dnk5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02692",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7220450465",
  "pdf_url": "https://arxiv.org/pdf/2610.02692",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bs485f9c5dnk5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02692",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02692"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Hilbert-Schmidt Norms of Self-Commutator and Cross-Commutator on $N_η$-type Quotient Modules in Polydisc",
  "title_text": "Hilbert-Schmidt Norms of Self-Commutator and Cross-Commutator on N_η-type Quotient Modules in Polydisc",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/bs485f9c5dnk5",
  "venue": "arXiv",
  "versions": []
}