Quadratic Lyapunov functions for linear skew-product flows and weighted composition operators

Abstract

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 C0-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. © 1995, Khayyam Publishing.

Publication Title

Differential and Integral Equations

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