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

AbstractIn 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∈L(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 equationhyperinvariant subspaceoperator matrices2020 Mathematics Subject Classifications: Primary 15A2447A0847A1047A1547A62 AcknowledgmentsThe authors wish to thank the referees for their invaluable comments on the original draft.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThe 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].