On the ⋆-Sylvester equation AX ± X⋆ B⋆ = C

We consider the solution of the ?-Sylvester equation AX±X?B? = C, for ? = T,H and A,B,∈ Cm×n, and some related linear matrix equations (AXB? ± X? = C, AXB? ± CX?D? = E, AX ± X?A? = C, AX ± Y B = C, AXB ± CY D = E, AXA? ± BY B? = C and AXB ± (AXB)? = C). Solvability conditions and stable numerical methods are considered, in terms of the (generalized and periodic) Schur, QR and (generalized) singular value decompositions. We emphasize on the square cases where m = n but the rectangular cases will be considered. The ?-sylvester equation is important in the solution of some generalized algebraic Riccati equations by Newton’s method.