Solving the Sylvester equation in Banach modules
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911680417497088 |
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| author | Djordjević, Bogdan |
| author_facet | Djordjević, Bogdan |
| contents | For given unital complex Banach algebras $\mathcal{A}_1$ and $\mathcal{A}_2$, let $\mathfrak{M}$ be a Banach module acting between them. Let $a\in \mathcal{A}_1$, $b\in\mathcal{A}_2$, and $c\in\mathfrak{M}$ be provided such that $σ_{\mathcal{A}_1}(a)\capσ_{\mathcal{A}_2}(b) \neq\emptyset$. In this paper we completely characterize the consistency of the Sylvester equation $$ax-xb=c.$$ Precisely, we establish verifiable sufficient and necessary solvability conditions, and we provide some formulas for particular solutions $x\in\mathfrak{M}$ when the equation is solvable. Moreover, we characterize the uniqueness of the solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_13419 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Solving the Sylvester equation in Banach modules Djordjević, Bogdan Functional Analysis Operator Algebras 46H25, 46H10, 46H30, 47D03, 47A08 For given unital complex Banach algebras $\mathcal{A}_1$ and $\mathcal{A}_2$, let $\mathfrak{M}$ be a Banach module acting between them. Let $a\in \mathcal{A}_1$, $b\in\mathcal{A}_2$, and $c\in\mathfrak{M}$ be provided such that $σ_{\mathcal{A}_1}(a)\capσ_{\mathcal{A}_2}(b) \neq\emptyset$. In this paper we completely characterize the consistency of the Sylvester equation $$ax-xb=c.$$ Precisely, we establish verifiable sufficient and necessary solvability conditions, and we provide some formulas for particular solutions $x\in\mathfrak{M}$ when the equation is solvable. Moreover, we characterize the uniqueness of the solutions. |
| title | Solving the Sylvester equation in Banach modules |
| topic | Functional Analysis Operator Algebras 46H25, 46H10, 46H30, 47D03, 47A08 |
| url | https://arxiv.org/abs/2605.13419 |