Evolutionary semigroups and Lyapunov theorems in Banach spaces

Let us consider an autonomous differential equation v ′ = Av in a Banach space E, where A is a generator of C0-semigroup {e tA}t≥0. Denote, as usual, s(A) = sup{Re λ: λ ∈ σ(A)} and ω(A) = inf{ω: �e tA � ≤ Me ωt}