Admissible Banach function spaces for linear dynamics with nonuniform behavior on the half-line

For nonuniform exponentially bounded evolution families on the half-line, we introduce a special class of Banach function spaces, on which we define certain $$C_{0}$$ -semigroups. We characterize the existence of nonuniform exponential stability in terms of invertibility of the corresponding infinitesimal generators. The invertibility of these generators is connected to a particular type of admissible exponents that are specific to nonuniform behavior. For the bounded orbits, nonuniform exponential stability results from a spectral property of generators. The $$C_{0}$$ -semigroups we deal with verify the spectral mapping theorem, as well as the evolution semigroups, in the uniform case. In particular, our results directly apply to all linear differential equations with finite Lyapunov exponent.