Solutions of Sylvester equation in $C^*$-modular operators
UDC 517.9 We study the solvability of the Sylvester equation $AX + Y B = C$ and the operator equation $AXD + FY B = C$ in the general setting of the adjointable operators between Hilbert $C^*$ -modules. Based on the Moore – Penrose inverses of the associated operators, we propose necessary and suffi...
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| Date: | 2021 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2021
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/152 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | UDC 517.9
We study the solvability of the Sylvester equation $AX + Y B = C$ and the operator equation $AXD + FY B = C$ in the general setting of the adjointable operators between Hilbert $C^*$ -modules. Based on the Moore – Penrose inverses of the associated operators, we propose necessary and sufficient conditions for the existence of solutions to these equations, and obtain the general expressions of the solutions in the solvable cases. We also provide an approach to the study of the positive solutions for a special case of Lyapunov equation. |
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| DOI: | 10.37863/umzh.v73i3.152 |