On the generalized Sylvester operator equation AX−Y B = C

Abstract

In this paper, we study the necessary and sufficient conditions to have a solution pair $(X,Y)$ to the generalized Sylvester equation AX−Y B = C where $A,B,C\in \mathcal{L}(\mathcal{H})$ are given. Moreover, we give some connections the generalized Fuglede-Putnam property and the solution pair $(X,Y)$ of the operator equation AX−Y B = C. Finally, we consider some properties of the operator equation AX−Y B = C when A and B belong to the special classes of operators.

Keywords:

• Sylvester operator equation
• hyperinvariant subspace
• operator matrices

2020 Mathematics Subject Classifications:

• Primary 15A24
• 47A08
• 47A10
• 47A15
• 47A62

Acknowledgments

The authors wish to thank the referees for their invaluable comments on the original draft.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was supported by the National Research Foundation of Korea (NRF) funded by Ministry of Science and ICT [grant number NRF-2020R1A2C1A01006036]. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education [grant number 2019R1A6A1A11051177] and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) [grant number 2019R1F1A1058633]. The third author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) [grant number 2022R1H1A2091052].