Exponentially Dichotomous Operators and Exponential Dichotomy of Evolution Equations on the Half-Line

Abstract.

To an evolution family on the half-line $\mathcal{U} = (U(t, s))_{t\geq s\geq 0}$ of bounded operators on a Banach space X we associate operators I _X and I _Z related to the integral equation $u(t) = U(t, s)u(s) + \int^{t}_{s} U(t, \xi) f (\xi)d\xi$ and a closed subspace Z of X. We characterize the exponential dichotomy of $\mathcal{U}$ by the exponential dichotomy and the quasi-exponential dichotomy of the operators X we associate operators I _X and I _Z , respectively.