Let U = (U(t, s))t?s?0 be an evolution family on the half-line of bounded linear operators on a
Banach space X. We introduce operators G0, GX and IX on certain spaces of X-valued continuous functions
connected with the integral equation u(t) = U(t, s)u(s) + ?t
s U(t, ??)f/(??)d??, and we characterize
exponential stability, exponential expansiveness and exponential dichotomy of U by properties of G0, GX
and IX, respectively. This extends related results known for finite dimensional spaces and for evolution
families on the whole line, respectively.
