Uniqueness of solution of a generalized ⋆-Sylvester matrix equation

We present necessary and sufficient conditions for the existence of a unique solution of the generalized ⋆-Sylvester matrix equation AXB+CX⋆D=E, where A,B,C,D,E are square matrices of the same size with real or complex entries, and where ⋆ stands for either the transpose or the conjugate transpose. This generalizes several previous uniqueness results for specific equations like the ⋆-Sylvester or the ⋆-Stein equations.