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

We analyze the consistency and uniqueness of solution of the generalized (Formula presented.) -Sylvester equation (Formula presented.), with (Formula presented.), and (Formula presented.) being complex matrices (and (Formula presented.) 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 (Formula presented.) and (Formula presented.), respectively. This approach deals with matrices whose size is of the same order as that of (Formula presented.), and (Formula presented.), 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 (Formula presented.), and (Formula presented.) 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 (Formula presented.) -Sylvester equation (Formula presented.) and the (Formula presented.) -Stein equation (Formula presented.) to have at most one solution or to be consistent, for any right-hand side (Formula presented.).