Solution estimates for the discrete lyapunov equation in a hilbert space and applications to difference equations

The paper is devoted to the discrete Lyapunov equation X - A*XA = C, where A and C are given operators in a Hilbert space H and X should be found. We derive norm estimates for solutions of that equation in the case of unstable operator A, as well as refine the previously-published estimates for the equation with a stable operator. By the point estimates, we establish explicit conditions, under which a linear nonautonomous difference equation in H is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in H. Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators.