Exponential and Polynomial Dichotomies of Operator Semigroups on Banach Spaces

Let A generate a C0–semigroup T (·) on a Banach space X such that the resolvent R(iτ, A) exists and is uniformly bounded for τ ∈ R. We show that there exists a closed, possibly unbounded projection P on X commuting with T (t). Moreover, T (t)x decays exponentially as t → ∞ for x in the range of P and T (t)x exists and decays exponentially as t→ −∞ for x in kernel of P . The domain of P depends on the Fourier type of X. If R(iτ, A) is only polynomially bounded, one obtains a similar result with polynomial decay. As an application we study a partial functional differential equation.