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, 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) + ∫^t_s U(t, ξ)f/(ξ)dξ, and we characterize exponential stability, exponential expansiveness and exponential dichotomy of U by properties of G₀, G_X and I_X, respectively. This extends related results known for finite dimensional spaces and for evolution families on the whole line, respectively.

Van Minh N., Rabiger F., Schnaubelt R.

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