Octonionic Para-linear Self-Adjoint Operators and Spectral Decomposition

Abstract: This paper presents a groundbreaking advancement in the theory of operators defined on octonionic Hilbert spaces, successfully resolving a fundamental challenge that has persisted for over six decades. Due to the intrinsic non-associative nature of octonions, conventional linear operator theory encounters profound structural difficulties. We make use of an original conceptual framework termed para-linearity, an innovative generalization of linearity that naturally accommodates the octonionic algebraic structure. Within this newly established paradigm, we systematically develop an appropriate algebraic setting by defining a carefully designed operator algebra and an adjoint operation which, together, recapture essential analytic properties previously inaccessible in this context. We identify a geometric structure, the slice cone, as the fundamental object encoding spectral properties typically derived through sesquilinear forms. We obtain a rigorous characterization of self-adjointness which indicates how to introduce a new notion of strong eigenvalues. For every compact, para-linear, self-adjoint operator with strong eigenvalues, we can establish the spectral decomposition theorem and functional calculi.