Parole chiave

• Riesz-Dunford functional calculus
• octonions
• resolvent operator
• octonionic spectrum
• slice regular functional calculus

Abstract The Riesz-Dunford functional calculus over the algebra of octonions has long been an open problem due to the nonassociativity of octonions. Two core obstacles hinder its development: first, generalizing the resolvent operator series identity produces unexpected associator terms that invalidate standard expansions; second, nonassociativity spoils the analyticity of the resolvent operator, a key property for defining a functional calculus via Cauchy integrals. In this talk, we introduce the Riesz-Dunford functional calculus for bounded power-associative para-linear operators in Banach octonionic bimodules. To address these issues, we introduce several pivotal concepts: power-associative operators, which eliminate the unwanted associator terms and recover valid resolvent series expansions; the regular inverse of $R_s-T$, for $s\in\mathbb{O}$, which serves as the octonionic analogue of the resolvent operator; and $\mathbb{C}_J$-extendable and $\mathbb{C}_J$-liftable power-associative operators, which characterize the slice regularity of the resolvent operators. Based on these notions, we define two types of octonionic spectra: the pull-back spectrum $\sigma^\ast(T)$ and the push-forward spectrum $\sigma_\ast(T)$. These give rise to the left and right slice regular functional calculi of bounded power-associative para-linear operators, respectively.