Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line

Let U = (U(t, s))t greater than or equal to s greater than or equal to o be an evolution family on the half-line of bounded linear operators on a Bana ch space X. We introduce operators G(o), G(x) and I-x on certain spaces of X-valued continuous functions connected with the integral equation u(t) = U (t, s)u(s) + integral(s)(t) U(t, xi) f (xi) d xi, and we characterize expon ential stability, exponential expansiveness and exponential dichotomy of U by properties of G(o), G(x) and I-x respectively. This extends related resu lts known for finite dimensional spaces and for evolution families on the w hole line, respectively.