Octonionic Hilbert Spaces and Para-linear Operators

The theory of octonionic Hilbert spaces has advanced significantly with the introduction of octonionic para-linearity. This article examines recent developments in para-linear operators, their duals, and self-adjoint para-linear operators. A key distinction is that octonionic matrices lack octonionic linearity unless they are real matrices, affecting their applications in Albert algebras and theoretical physics. To address the non-associative nature of octonions, we introduce fractional subspaces to modify classical relations between the kernel and range. Notable results include a new understanding of dual operations via the Riesz representation theorem and the Jordan decomposition for self-adjoint para-linear operators of finite rank, closely linked to the slice cone of octonionic Hilbert spaces.