Invertibility and dichotomy of differential operators on a half-line

A linear ordinary differential operator with bounded coefficients satisfying certain homogeneous initial conditions is shown to be invertible on L_n²(0, ∞) if and only if the underlying system of differential equations has a dichotomy. Moreover, in that case the operator is proved to be a direct sum of two infinitesimal generators of C₀-semigroups, one of which has support on the negative half-line and the other on the positive half-line. The effect of perturbations of the initial values on the dichotomy is also described.