A Compactness Criterion for Non-Self-Adjoint Spectral Decompositions
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| Natura: | Recurso digital |
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2025
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| _version_ | 1866901746515705856 |
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| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | This paper develops a novel compactness criterion for the spectral decomposition of non-self-adjoint operators acting on complex Hilbert spaces. Unlike self-adjoint operators, which guarantee real eigenvalues and orthogonal eigenbases, non-self-adjoint operators present significant challenges due to potentially complex spectra, lack of orthogonal eigenvectors, and the presence of generalized eigenvectors. We propose a condition based on the asymptotic behavior of the resolvent operator in conjunction with properties of certain operator ideals, specifically focusing on how deviations from normality impact the discreteness and structure of the spectrum. The criterion establishes sufficient conditions under which a non-self-adjoint operator admits a Riesz basis of generalized eigenvectors, effectively extending the concept of spectral decomposition to a broader class of operators than traditionally covered by compact perturbations or specific normal forms. Our approach leverages techniques from functional analysis, perturbation theory, and the theory of operator semigroups, providing a more refined understanding of when spectral representations remain viable for operators with non-trivial essential spectra. This work aims to bridge the gap between abstract spectral theory and its applicability in quantum mechanics, control theory, and other areas where non-self-adjoint dynamics are prevalent. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17804415 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Compactness Criterion for Non-Self-Adjoint Spectral Decompositions Revista, Zen MATH, 10 This paper develops a novel compactness criterion for the spectral decomposition of non-self-adjoint operators acting on complex Hilbert spaces. Unlike self-adjoint operators, which guarantee real eigenvalues and orthogonal eigenbases, non-self-adjoint operators present significant challenges due to potentially complex spectra, lack of orthogonal eigenvectors, and the presence of generalized eigenvectors. We propose a condition based on the asymptotic behavior of the resolvent operator in conjunction with properties of certain operator ideals, specifically focusing on how deviations from normality impact the discreteness and structure of the spectrum. The criterion establishes sufficient conditions under which a non-self-adjoint operator admits a Riesz basis of generalized eigenvectors, effectively extending the concept of spectral decomposition to a broader class of operators than traditionally covered by compact perturbations or specific normal forms. Our approach leverages techniques from functional analysis, perturbation theory, and the theory of operator semigroups, providing a more refined understanding of when spectral representations remain viable for operators with non-trivial essential spectra. This work aims to bridge the gap between abstract spectral theory and its applicability in quantum mechanics, control theory, and other areas where non-self-adjoint dynamics are prevalent. |
| title | A Compactness Criterion for Non-Self-Adjoint Spectral Decompositions |
| url | https://doi.org/10.5281/zenodo.17804415 |