The Spectral Radius, Hyperbolic Operators and Lyapunov’s Theorem

Let A be the generator of a C ₀-semigroup T(t), t ≥ 0, on a Hilbert space H. It is the well known Lyapunov Theorem that T(t) is exponentially stable if and only if for some (or for any) positive definite operator Q, the operator equation AX + XA* = −Q has a positive definite solution, see [4,5]. Thus, there exists a close relationship between the asymptotic behaviour of C ₀-semigroups and the solutions of differential equations, from one side, and the solvability of the operator equations of the Lyapunov form, AX + XA* = −Q, from the other side.