  We propose a novel operator-theoretic framework to study global stability of
nonlinear systems. Based on the spectral properties of the so-called Koopman
operator, our approach can be regarded as a natural extension of classic linear
stability analysis to nonlinear systems. The main results establish the
(necessary and sufficient) relationship between the existence of specific
eigenfunctions of the Koopman operator and the global stability property of
hyperbolic fixed points and limit cycles. These results are complemented with
numerical methods which are used to estimate the region of attraction of the
fixed point or to prove in a systematic way global stability of the attractor
within a given region of the state space.
