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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.11709 |
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| _version_ | 1866912158854414336 |
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| author | Holubová, Gabriela Nečesal, Petr |
| author_facet | Holubová, Gabriela Nečesal, Petr |
| contents | In this paper, we study the Fuč\'ık spectrum of a square matrix $A$ and provide necessary and sufficient conditions for the existence of Fuč\'ık curves emanating from the point $(λ,λ)$ with $λ$ being a real eigenvalue of $A$. We extend recent results by Maroncelli (2024) and remove his assumptions on symmetry of $A$ and simplicity of $λ$. We show that the number of Fuč\'ık curves can significantly exceed the multiplicity of $λ$ and determine all the possible directions they can emanate in. We also treat the situation when the algebraic multiplicity of $λ$ is greater than the geometric one and show that in such a case the Fuč\'ık curves can loose their smoothness and provide the slopes of their "one-sided tangent lines". Finally, we offer two possible generalizations: the situation off the diagonal and Fuč\'ık spectrum of a general Fredholm operator on the Hilbert space with a lattice structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11709 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fuč\'ık spectrum for discrete systems: curves and their tangent lines Holubová, Gabriela Nečesal, Petr Spectral Theory 15A18, 15A24, 15A60, 47A75 In this paper, we study the Fuč\'ık spectrum of a square matrix $A$ and provide necessary and sufficient conditions for the existence of Fuč\'ık curves emanating from the point $(λ,λ)$ with $λ$ being a real eigenvalue of $A$. We extend recent results by Maroncelli (2024) and remove his assumptions on symmetry of $A$ and simplicity of $λ$. We show that the number of Fuč\'ık curves can significantly exceed the multiplicity of $λ$ and determine all the possible directions they can emanate in. We also treat the situation when the algebraic multiplicity of $λ$ is greater than the geometric one and show that in such a case the Fuč\'ık curves can loose their smoothness and provide the slopes of their "one-sided tangent lines". Finally, we offer two possible generalizations: the situation off the diagonal and Fuč\'ık spectrum of a general Fredholm operator on the Hilbert space with a lattice structure. |
| title | Fuč\'ık spectrum for discrete systems: curves and their tangent lines |
| topic | Spectral Theory 15A18, 15A24, 15A60, 47A75 |
| url | https://arxiv.org/abs/2412.11709 |