Lyapunov regularity in Hilbert spaces

The regularity theory presented in Chapter 10 is closely related to the existence of nonuniform exponential dichotomies (see Section 10.2). Unfortunately, it can only be applied to dynamical systems in finite-dimensional spaces. Hence, it is important to develop counterparts of the theory in infinitedimensional spaces. The main goal of this chapter is precisely to introduce a version of Lyapunov regularity in Hilbert spaces, imitating as much as possible the classical theory introduced by Lyapunov in Rn. We also describe the geometric consequences of regularity, that are related to the existence of exponential growth rates of norms, angles, and volumes determined by the solutions. We shall see in Chapter 12 that this generalization can be used to establish the persistence of the asymptotic stability of solutions of nonlinear equations under sufficiently small perturbations of Lyapunov regular equations, again in the infinite-dimensional setting of Hilbert spaces. The exposition is based in [7].