Octonionic Riesz-Dunford functional calculus

@misc{huo2026octonionic,
  title = {{Octonionic Riesz-Dunford functional calculus}},
  author = {Qinghai Huo and Guangbin Ren and Irene Sabadini and Zhenghua Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.07183},
  eprint = {2605.07183},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.07183},
}

TY  - UNPB
TI  - Octonionic Riesz-Dunford functional calculus
AU  - Qinghai Huo
AU  - Guangbin Ren
AU  - Irene Sabadini
AU  - Zhenghua Xu
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2605.07183
UR  - https://arxiv.org/abs/2605.07183
AB  - The Riesz-Dunford functional calculus over the algebra of octonions, denoted by 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∈ Ø (which serve as the octonionic versions of the resolvent operator), ℂ_J-extendable power-associative operators, and 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.
ER  -