Method of similar operators in harmonious Banach spaces
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913293842513920 |
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| author | Baskakov, Anatoly G. Krishtal, Ilya A. Uskova, Natalia B. |
| author_facet | Baskakov, Anatoly G. Krishtal, Ilya A. Uskova, Natalia B. |
| contents | We consider similarity transformations of a perturbed linear operator $A-B$ in a complex Banach space $\mathcal{X}$, where the unperturbed operator $A$ is a generator of a Banach $L_1(\mathbb{R})$-module and the perturbation operator $B$ is a bounded linear operator. The result of the transformation is a simpler operator $A-B_0$. For example, if $A$ is a differentiation operator and $B$ is an operator of multiplication by an operator-valued function, then $B_0$ is an operator of multiplication by a function that is a restriction of an entire function of exponential type and could be $0$ in some cases. As a consequence, we derive the spectral invariance of the operator $A-B$ in a large class of spaces. The study is based on a widely applicable modification of the method of similar operators that is also presented in the paper. This non-traditional modification is rooted in the spectral theory of Banach $L_1(\mathbb{R})$-modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01227 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Method of similar operators in harmonious Banach spaces Baskakov, Anatoly G. Krishtal, Ilya A. Uskova, Natalia B. Functional Analysis Spectral Theory We consider similarity transformations of a perturbed linear operator $A-B$ in a complex Banach space $\mathcal{X}$, where the unperturbed operator $A$ is a generator of a Banach $L_1(\mathbb{R})$-module and the perturbation operator $B$ is a bounded linear operator. The result of the transformation is a simpler operator $A-B_0$. For example, if $A$ is a differentiation operator and $B$ is an operator of multiplication by an operator-valued function, then $B_0$ is an operator of multiplication by a function that is a restriction of an entire function of exponential type and could be $0$ in some cases. As a consequence, we derive the spectral invariance of the operator $A-B$ in a large class of spaces. The study is based on a widely applicable modification of the method of similar operators that is also presented in the paper. This non-traditional modification is rooted in the spectral theory of Banach $L_1(\mathbb{R})$-modules. |
| title | Method of similar operators in harmonious Banach spaces |
| topic | Functional Analysis Spectral Theory |
| url | https://arxiv.org/abs/2404.01227 |