{
  "abstract": "We consider right-linear operators $T$ on a right Hilbert module $V$ over the Clifford algebra R_n, whose S-spectrum lies in an acute double sector. For these bisectorial operators, we introduce spectral projectors P_\\pm associated with the two cones of the double sector. They decompose the Hilbert module into two submodules V=V_++V_-, and the bisectorial operator T into two sectorial operators T|_\\pm. A crucial but non-trivial, cornerstone in this theory is the boundedness of the projectors P_\\pm, which is, in turn connected to a bounded H^\\infty-functional calculus of the operator T. We provide two practical criteria: either the squared operator admits a bounded H^\\infty-functional calculus, or the operator is m-accretive. Finally, we apply these results to the gradient operator \\nabla_a with nonconstant coefficients. For the particular gradient with constant coefficients, we are even able to derive explicit representations of the submodules V_\\pm and the projectors P_\\pm in Fourier space. Moreover, we identify the sign of the gradient operator with the Clifford-Hilbert transform. This sign plays a central role in the fractional powers of vector operators, which are used, for instance, in the non-local Fourier law of heat propagation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26382",
  "author_total": 2,
  "authors": [
    {
      "name": "Francesco Mantovani"
    },
    {
      "name": "Peter Schlosser"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9w6negmvyt3kp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26382",
  "fulltext_url": null,
  "id": "9w6negmvyt3kp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26382",
  "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.SP"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7214128555",
  "pdf_url": "https://arxiv.org/pdf/2609.26382",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9w6negmvyt3kp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26382",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26382"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Spectral projectors of bisectorial Clifford operators and applications to the generalized gradient",
  "title_text": "Spectral projectors of bisectorial Clifford operators and applications to the generalized gradient",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/9w6negmvyt3kp",
  "venue": "arXiv",
  "versions": []
}