Algebras closed by $J$-hermitianity in displacement formulas

We introduce the notion of $J$-Hermitianity of a matrix, as a generalization of Hermitianity, and, more generally, of closure by $J$-Hermitianity of a set of matrices. Many well known algebras, like upper and lower triangular Toeplitz, Circulants and $\tau$ matrices, as well as certain algebras that have dimension higher than the matrix order, turn out to be closed by $J$-Hermitianity. As an application, we generalize some theorems about displacement decompositions presented in [1, 2], by assuming the matrix algebras involved closed by $J$-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.