Spectral inclusions of perturbed normal operators and applications
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866916337893244928 |
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| author | Moreno, Javier Winklmeier, Monika |
| author_facet | Moreno, Javier Winklmeier, Monika |
| contents | We consider a normal operator $T$ on a Hilbert space $H$. Under various assumptions on the spectrum of $T$, we give bounds for the spectrum of $T+A$ where $A$ is $T$-bounded with relative bound less than 1 but we do not assume that $A$ is symmetric or normal. If the imaginary part of the spectrum of $T$ is bounded, then the spectrum of $T+A$ is contained in the region between certain hyperbolas whose asymptotic slope depends on the $T$-bound of $A$. If the spectrum of $T$ is contained in a bisector, then the spectrum of $T+A$ is contained in the area between certain rotated hyperbola. The case of infinite gaps in the spectrum of $T$ is studied. Moreover, we prove a stability result for the essential spectrum of $T+A$. If $A$ is even $p$-subordinate to $T$, then we obtain stronger results for the localisation of the spectrum of $T+A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_12550 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spectral inclusions of perturbed normal operators and applications Moreno, Javier Winklmeier, Monika Spectral Theory 47A55, 47A10, 47B12, 47B28 We consider a normal operator $T$ on a Hilbert space $H$. Under various assumptions on the spectrum of $T$, we give bounds for the spectrum of $T+A$ where $A$ is $T$-bounded with relative bound less than 1 but we do not assume that $A$ is symmetric or normal. If the imaginary part of the spectrum of $T$ is bounded, then the spectrum of $T+A$ is contained in the region between certain hyperbolas whose asymptotic slope depends on the $T$-bound of $A$. If the spectrum of $T$ is contained in a bisector, then the spectrum of $T+A$ is contained in the area between certain rotated hyperbola. The case of infinite gaps in the spectrum of $T$ is studied. Moreover, we prove a stability result for the essential spectrum of $T+A$. If $A$ is even $p$-subordinate to $T$, then we obtain stronger results for the localisation of the spectrum of $T+A$. |
| title | Spectral inclusions of perturbed normal operators and applications |
| topic | Spectral Theory 47A55, 47A10, 47B12, 47B28 |
| url | https://arxiv.org/abs/2309.12550 |