Lyapunov type characterization of hyperbolic behavior

We give a complete characterization of the uniform hyperbolicity and nonuniform hyperbolicity of a cocycle with values in the space of bounded linear operators acting on a Hilbert space in terms of the existence of appropriate quadratic forms. Our work unifies and extends many results in the literature by considering the general case of not necessarily invertible cocycles acting on a Hilbert space over an arbitrary invertible dynamics. As a nontrivial application of, we study the persistence of hyperbolicity under small linear perturbations.