Algebras Closed by J-Hermitianity in Displacement Formulas

We introduce the notion of -Hermitianity of a matrix, as a generalization of Hermitianity, and, more generally, of closure by -Hermitianity of a set of matrices. Many well known algebras, like upper and lower triangular Toeplitz, Circulants and matrices, as well as certain algebras that have dimension higher than the matrix order, turn out to be closed by -Hermitianity. As an application, we generalize some theorems about displacement decompositions presented in [1, 2], by assuming the matrix algebras involved closed by -Hermitianity. Even if such hypothesis on the structure is not necessary in the case of algebras generated by one matrix, as it has been proved in [3], our result is relevant because it could yield new low complexity displacement formulas involving not one-matrix-generated commutative algebras.