The Daleckij-Krein method for constructing a quadratic Lyapunov function for the equation $f'=Df(t)$ in Hilbert space is extended to include the case of an unbounded operator $D$ that generates a $C_0$-group. The extension is applied to obtain a quadratic Lyapunov function for the case of a group of weighted composition operators generated by a flow on a compact metric space together with a cocycle over this flow. These results are used to characterize the hyperbolicity of linear skew-product flows in terms of the existence of such a Lyapunov function. Also, the "trajectorial" method for constructing the Lyapunov function is discussed. Interrelations with Schrödinger, Riccati and Hamiltonian equations are discussed and an application to geodesic flows on two-dimensional Riemannian manifolds is given.
"Quadratic Lyapunov functions for linear skew-product flows and weighted composition operators." Differential Integral Equations 8 (2) 289 - 307, 1995. https://doi.org/10.57262/die/1369083470