Integral characterizations for the dichotomy of evolution families (Q935548)

In this paper, some results by Datko, Pazy, Barbashin and others on the existence of uniform exponential dichotomies for evolution families \(\Phi\) on a Banach space \(X\) are generalized. Under the assumption that a dichotomy projector family exists, i.e., a strongly continuous projector-valued function \(P:[0,\infty)\to L(X)\) with \(\Phi(t,t_0)P(t_0)=P(t)\Phi(t,t_0)\) and the property that \(\Phi(t,t_0):\mathrm{Ker}\,(P(t_0))\to\mathrm{Ker}\,(P(t))\) is an isomorphism, the authors give necessary and sufficient integral conditions under which the evolution family possesses a uniform exponential dichotomy with projections \(P\).