Solvability and uniqueness of solution of generalized ★ $\star$ -Sylvester equations with arbitrary coefficients

We analyze the consistency and uniqueness of solution of the generalized $★$-Sylvester equation $AXB+C{X}^{★}D=E$, with $A,B,C,D$, and $E$ being complex matrices (and $★$ being either the transpose or the conjugate transpose). In particular, we obtain characterizations for the equation to have at most one solution and to be consistent for any right-hand side. Such characterizations are given in terms of spectral properties of the matrix pencils and , respectively. This approach deals with matrices whose size is of the same order as that of $A,B,C$, and $D$, contrary to the naive procedure that addresses the equation as a linear system, whose coefficient matrix can be much larger. The characterizations are valid in the most general setting, namely for all coefficient matrices $A,B,C$, and $D$ for which the equation is well-defined, and generalize the known characterizations for the case where they are all square. As a corollary, we obtain necessary and sufficient conditions for the $★$-Sylvester equation $AX+X^{★}D=E$ and the $★$-Stein equation $X+C{X}^{★}D=E$ to have at most one solution or to be consistent, for any right-hand side $E$.