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

We consider the solution of the ★-Sylvester equations AX ± X★B★ = C, for ★ = T, H   and A,B,∈Cn×nA,B,∈Cn×n, and the related linear matrix equations AXB★ ± X★ = C, AXB★ ± CX★D★ = E and AX ± X★A★ = C. Solvability conditions and numerical methods are considered, in terms of the (generalized and periodic) Schur and QR decompositions. We emphasize the square cases where m = n but the rectangular cases will be considered.