On the spectrum of evolution operators generated by hyperbolic systems (Q1076898)

We prove that the asymptotic behavior of semi-groups generated by mixed- initial value problems for hyperbolic systems in one space variable is determined by the zeros of the characteristic equation. The proof is based on the following reduction principle: If T(t) and \(T_ 0(t)\) are semigroups whose difference \(T(t)-T_ 0(t)\) is compact and the asymptotic behavior of \(T_ 0(t)\) is determined by the spectrum of its infinitesimal generator then the same thing happens to T(t). We construct a reduced system (which is much simpler than the given one) in order to be able to apply this principle. The construction of the reduced system is presented in the nonautonomous case because it has applications in the perturbation theory for linear systems which are time periodic.