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A complement to spectral mapping theorems for C0‐semigroups

We obtain a complement to the classical spectral mapping (inclusion) theorems for C0‐semigroups. More precisely, we show that if (T(t))t⩾0 is a C0‐semigroup on a complex Banach space X with generator A and if, for fixed k∈ ℕ and t > 0, the operator (T(t)−I)k has closed range, then the range of Ak is also closed. In particular, for k = 1 this implies the validity of the following version of the spectral mapping (inclusion) theorem etσℛ(A) ⊂ σℛ (T(t)),t > 0, where σℛ(A) := {λ∈ℂ: the range of λ−A is not closed}. Our results are, in a sense, optimal.