On Sylvester equations in Banach subalgebras
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909835766792192 |
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| author | Fang, Qiquan Shin, Chang Eon Sun, Qiyu |
| author_facet | Fang, Qiquan Shin, Chang Eon Sun, Qiyu |
| contents | Let ${\mathcal B}$ be a Banach algebra and ${\mathcal A}$ be a Banach subalgebra that admits norm-controlled inversion in ${\mathcal B}$. In this work, we take $A, B$ in the Banach subalgebra ${\mathcal A}$ with their spectra in the Banach algebra ${\mathcal B}$ being disjoint, and show that the operator Sylvester equation $ BX-XA=Q$ has a unique solution $X\in {\mathcal A}$ for every $Q\in {\mathcal A}$. Under the additional assumptions that ${\mathcal B}$ is the operator algebra ${\mathcal B}(H)$ on a Hilbert space $H$ and that $A$ and $B$ are normal in ${\mathcal B}(H)$, an explicit norm estimate for the solution $X$ of the above operator Sylvester equation is provided in this work. In addition, the above conclusion on norm control is applied to Banach subalgebras of localized infinite matrices and integral operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09752 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Sylvester equations in Banach subalgebras Fang, Qiquan Shin, Chang Eon Sun, Qiyu Functional Analysis 47G10, 47B38, 47A10, 47B49 Let ${\mathcal B}$ be a Banach algebra and ${\mathcal A}$ be a Banach subalgebra that admits norm-controlled inversion in ${\mathcal B}$. In this work, we take $A, B$ in the Banach subalgebra ${\mathcal A}$ with their spectra in the Banach algebra ${\mathcal B}$ being disjoint, and show that the operator Sylvester equation $ BX-XA=Q$ has a unique solution $X\in {\mathcal A}$ for every $Q\in {\mathcal A}$. Under the additional assumptions that ${\mathcal B}$ is the operator algebra ${\mathcal B}(H)$ on a Hilbert space $H$ and that $A$ and $B$ are normal in ${\mathcal B}(H)$, an explicit norm estimate for the solution $X$ of the above operator Sylvester equation is provided in this work. In addition, the above conclusion on norm control is applied to Banach subalgebras of localized infinite matrices and integral operators. |
| title | On Sylvester equations in Banach subalgebras |
| topic | Functional Analysis 47G10, 47B38, 47A10, 47B49 |
| url | https://arxiv.org/abs/2407.09752 |