Octonionic Riesz-Dunford functional calculus

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 $s\in \O$ (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 $\sigma^*(T)$ and the push-forward spectrum $\sigma_*(T)$. 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.