Simultaneous solutions of Sylvester equations and idempotent matrices separating the joint spectrum

We investigate simultaneous solutions of the matrix Sylvester equations AiX-XBi=Ci,i=1,2,…,k, where {A1,…,Ak} and {B1,…,Bk} are k-tuples of commuting matrices of order m×m and p×p, respectively. We show that the matrix Sylvester equations have a unique solution X for every compatible k-tuple of m×p matrices {C1,…,Ck} if and only if the joint spectra σ(A1,…,Ak) and σ(B1,…,Bk) are disjoint. We discuss the connection between the simultaneous solutions of Sylvester equations and related questions about idempotent matrices separating disjoint subsets of the joint spectrum, spectral mapping for the differences of commuting k-tuples, and a characterization of the joint spectrum via simultaneous solutions of systems of linear equations.