Non-Hermitian Matrices of Even Order and Neutral Subspaces of Half the Dimension

Consider the sesquilinear matrix equation $X^*DX+AX+X^*B+C=0$, where all the matrices are square and have the same order $n$. With this equation, we associate a block matrix $M$ of double order $2n$. The solvability of the above equation turns out to be related to the existence of $n$-dimensional neutral subspaces for the matrix $M$. We indicate sufficiently general conditions ensuring the existence of such subspaces.