{
  "abstract": "The Riesz-Dunford functional calculus over the algebra of octonions, denoted by $\\mathbb{O}$, has long been an open problem due to the nonassociativity of octonions. Two core obstacles hinder its development: first, the generalization of the resolvent operator series identity produces unexpected associator terms that invalidate standard expansions; second, the nonassociativity spoils the analyticity of the resolvent operator, a key property for defining a functional calculus via Cauchy integrals. In this paper, we initiate the study of the Riesz-Dunford functional calculus for bounded power-associative para-linear operators in Banach octonionic bimodules. To address the above issues, we introduce several pivotal concepts: power-associative operators (to eliminate the unwanted associator terms and recover valid resolvent series expansions), the notions of regular inverse of $R_s-T$ for $s\\in Ø$ (which serve as the octonionic versions of the resolvent operator), $\\mathbb{C}_J$-extendable power-associative operators, and $\\mathbb{C}_J$-liftable power-associative operators (to characterize the slice regularity of the resolvent operators). Based on these notions, we define two types of octonionic spectra: the pull-back spectrum $σ^*(T)$ and the push-forward spectrum $σ_*(T)$. These give rise to the left and right slice regular functional calculi of bounded power-associative para-linear operators, respectively. This theory unifies the Riesz-Dunford functional calculus over division algebras ($ \\mathbb{C}, \\mathbb{H}, \\mathbb{O}$) and fills the six-decade-long gap in octonionic (nonassociative) functional analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.07183",
  "author_total": 4,
  "authors": [
    {
      "name": "Qinghai Huo"
    },
    {
      "name": "Guangbin Ren"
    },
    {
      "name": "Irene Sabadini"
    },
    {
      "name": "Zhenghua Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ftbqgrw8pbgak.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.07183",
  "fulltext_url": null,
  "id": "ftbqgrw8pbgak",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.07183",
  "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"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7160833655",
  "pdf_url": "https://arxiv.org/pdf/2605.07183",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ftbqgrw8pbgak.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2605.07183",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.07183"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Octonionic Riesz-Dunford functional calculus",
  "title_text": "Octonionic Riesz-Dunford functional calculus",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/ftbqgrw8pbgak",
  "venue": "arXiv",
  "versions": []
}