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Octonionic Para-linear Self-Adjoint Operators and Spectral Decomposition

Qinghai Huo, Guangbin Ren, Irene Sabadini

TL;DR

This work resolves a long-standing barrier in non-associative functional analysis by establishing a complete spectral theory for self-adjoint operators on octonionic Hilbert spaces through para-linearity. It introduces regular composition to form an involutive octonionic Banach algebra of para-linear operators, defines a corrected adjoint, and proves an octonionic polarization identity using the slice cone to characterize self-adjointness. The main results include a Hilbert-Schmidt-type spectral decomposition for compact self-adjoint para-linear operators, a geometric interpretation via slice projections, and a left/right functional calculus that is independent of spectral decomposition. The framework converts non-associativity from an obstruction to a structured tool, with anticipated impact on non-associative geometry, G2-manifold analysis, and non-associative quantum theories.

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.

Octonionic Para-linear Self-Adjoint Operators and Spectral Decomposition

TL;DR

This work resolves a long-standing barrier in non-associative functional analysis by establishing a complete spectral theory for self-adjoint operators on octonionic Hilbert spaces through para-linearity. It introduces regular composition to form an involutive octonionic Banach algebra of para-linear operators, defines a corrected adjoint, and proves an octonionic polarization identity using the slice cone to characterize self-adjointness. The main results include a Hilbert-Schmidt-type spectral decomposition for compact self-adjoint para-linear operators, a geometric interpretation via slice projections, and a left/right functional calculus that is independent of spectral decomposition. The framework converts non-associativity from an obstruction to a structured tool, with anticipated impact on non-associative geometry, G2-manifold analysis, and non-associative quantum theories.

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.

Paper Structure

This paper contains 44 sections, 39 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Motivations from Physics:
  3. Mathematical Obstacles:
  4. Para-Linearity: A Geometric Resolution.
  5. Principal Contributions.
  6. Impact and Applications.
  7. Algebraic Foundations of Para-Linear Operators
  8. Octonionic algebra
  9. Octonionic Module Theory
  10. Para-Linear Maps
  11. Hilbert $\mathbb{O}$-Bimodules
  12. Properties of Left Multiplications and Real Part
  13. Regular Composition: Resolving Non-Associative Obstructions
  14. Adjoint Operators via Regular Composition
  15. Adjoint Operator Definition
  16. ...and 29 more sections

Key Result

Proposition 2.3

For an $\mathbb{O}$-bimodule $M$, the real part$\operatorname{Re} M$ is defined as The projection $\operatorname{Re}: M \to \operatorname{Re} M$ satisfies:

Theorems & Definitions (94)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: Real Part Operator
  • Definition 2.4: Right Para-Linear Map
  • Theorem 2.5: Characterization of para-linearity
  • Lemma 2.6: Uniqueness Lemma huoqinghai2020nonass
  • Theorem 2.7: $\mathbb{O}$-Bimodule Structure huoqinghai2020nonass
  • Definition 2.8: Pre-Hilbert Right $\mathbb{O}$-Module huoqinghai2022Riesz
  • Lemma 2.9: Inner Product Identities
  • Theorem 2.10: Riesz Representation Theorem huoqinghai2022Riesz
  • ...and 84 more