Octonionic Para-linear Self-Adjoint Operators and Spectral Decomposition
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| Format: | Preprint |
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2025
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| _version_ | 1866911301935038464 |
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| author | Huo, Qinghai Ren, Guangbin Sabadini, Irene |
| author_facet | Huo, Qinghai Ren, Guangbin Sabadini, Irene |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_04707 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Octonionic Para-linear Self-Adjoint Operators and Spectral Decomposition Huo, Qinghai Ren, Guangbin Sabadini, Irene Functional Analysis Complex Variables 17A35, 46S10, Secondary 47B37, 47A70 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. |
| title | Octonionic Para-linear Self-Adjoint Operators and Spectral Decomposition |
| topic | Functional Analysis Complex Variables 17A35, 46S10, Secondary 47B37, 47A70 |
| url | https://arxiv.org/abs/2512.04707 |