Inequalities and Asymptotics for Riccati Matrix Difference Operators

In the first part inequalities for solutions of Riccati matrix difference equations are obtained which correspond to the linear Hamiltonian difference system D XksAkXkq1qBkUk, DUksCkXkq1yATkUk, where Ak, Bk, Ck, Xk, Uk are n=n-matrices with symmetric Bk and Ck. If the matrices Xkare invertible, then the matrices QksUkXyk 1 solve the Riccati matrix difference equation Qkq1sCkq IyATk.Qk IqBkQk.y1 IyAk.. In contrast to some recent papers dealing with these equations we do not assume that the matrices Bk are invertible. The second part of the paper deals with the asymptotic behaviour of solutions Qk l., as