An iterative method to solve Lyapunov equations

We present here a new splitting method to solve Lyapunov equations of the type $AP+P{A}^T=-B{B}^T$ in a Kronecker product form. Although that resulting matrix is of order $n^2$, each iteration of the method demands only two operations with the matrix $A$: a multiplication of the form $(A-\sigma I)\widehat{B}$ and a inversion of the form $(A-\sigma I{)}^{-1}\widehat{B}$. 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, which means that it should converge without depending on the starting vector. Nevertheless we present a theorem that enables us how to get a good starting vector for the method.