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 $\mathcal{H}$ 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 $\mathcal{H}$ is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in $\mathcal{H}$ . Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators. View Full-Text