{
  "abstract": "Given Hilbert space commuting operators $T, S \\in \\mcl(H)$, such that $T$ is $w$-hyponormal with $\\ker T \\subseteq \\ker T^*$ and $S$ is normal operator. Let $ϕ_{T, S} \\in \\mcl(\\mcl(H))$ be the elementary operator defined by $ϕ_{T, S} (X) = T X S^*-S X T^*$. In this paper, we show firstly that (1) $\\ker ϕ_{T, S} \\subset \\ker ϕ_{T^*, S^*}$. (2) The range of $ϕ_{T, S}$ is orthogonal to the kernel of $ϕ_{T, S}$ ( $ \\mcr(ϕ_{T, S}) \\perp \\ker ϕ_{T, S} $ ) if and only if $\\ker T \\cap \\ker S=\\{0\\}$. Secondly, we will extend these results to the elementary operator $Φ\\in \\mcl(\\mcl(H))$ defined by $\\;Φ(X)=A X D-C X B$ where $[A, C]=[B, D]= 0$. Related orthogonality results for the elementary operator $Φ$ are also given.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.18125",
  "author_total": 2,
  "authors": [
    {
      "name": "M. Morjane"
    },
    {
      "name": "M. Ech-chad"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cb3ws9eh3ps0c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.18125",
  "fulltext_url": null,
  "id": "cb3ws9eh3ps0c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.18125",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7203919348",
  "pdf_url": "https://arxiv.org/pdf/2608.18125",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cb3ws9eh3ps0c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2608.18125",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.18125"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Putnam-Fuglede commutativity and the range-kernel orthogonality of an elementary operator",
  "title_text": "Putnam-Fuglede commutativity and the range-kernel orthogonality of an elementary operator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/cb3ws9eh3ps0c",
  "venue": "arXiv",
  "versions": []
}