Groups of Weighted Translation Operators in Lp and Linear Nonautonomous Differential Equations

This paper is concerned with linear nonautonomous differential equations of the form x′ = A(t)x, defined on the real line. The main result gives necessary and sufficient conditions for the underlying equation to have an exponential dichotomy, in terms of spectral properties of the differential operator −d/dt + A(t) and the one-parameter semigroup of linear operators generated by it as operators acting in the function spaces Lp (1 ≤ p < ∞). The obtained results are a generalization for the case of Lp of those in a previous paper by the second author