Duality of invariant norms of continuous-time dynamical systems

In the study of the joint spectral radius of linear operators, an important role is played by invariant norms and invariant convex bodies. It is known that they are dual to each other in the sense of the polar transform and conjugation of operators. This property is used in studying the highest growth rate of trajectories of discrete-time linear dynamical systems. In the case of continuous-time systems, an analog of joint spectral radius is Lyapunov exponent. For such systems, invariant norms have been defined and their existence has been proved. However, there are no results concerning the invariant bodies of continuous-time systems in the literature. This paper is devoted to the duality of continuous-time systems. It is shown that there exists no natural notion of an invariant body for such systems and, therefore, the duality of systems is defined in terms of invariant norms.