An iterative method to solve Lyapunov equations

We present here a new splitting method to solve Lyapunov equations in a Kronecker product form. Although this resulting matrix is of order $n^2$, each iteration demands two operations with the matrix $A$: a multiplication of the form $(A-\sigma I) \tilde{B}$ and a inversion of the form $(A-\sigma I)^{-1}\tilde{B}$. We see that for some choice of a parameter the iteration matrix is such that all their eigenvalues are in absolute value less than 1. Moreover we present a theorem that enables us to get a good starting vector for the method.