Octonionic Riesz-Dunford functional calculus

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, 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 talk, we introduce 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 \(R_s-T\) (which serve as the 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.