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In [Q. Xu et al., The solutions to some operator equations, Linear AlgebraAppl.(2008), doi:10.1016/j.laa.2008.05.034], Xu et al. provided the necessaryand sufficient conditions for the existence of a solution to the equation$AXB^*-BX^*A^*=C$ in the general setting of the adjointable operators betweenHilbert $C^*$-modules. Based on the generalized inverses, they also obtainedthe general expression of the solution in the solvable case. In this paper, wegeneralize their work in the more general setting of ring $R$ with involution *and reobtain results for rectangular matrices and operators between Hilbert$C^*$-modules by embedding the "rectangles" into rings of square matrices orrings of operators acting on the same space.

General solutions to equation $axb^*-bx^*a^*=c$ in rings with involution

General solutions to equation $axb^-bx^a^*=c$ in rings with involution