Asymptotics of Analytic Semigroups, II

Let T(·) be an analytic C₀-semigroup of operators in a sector S_θ, such that ∥T(·)∥ is bounded in each proper subsector S_θ0. Let A be its generator, and let D ^∞(A) be its set of C^∞-vectors. It is observed that the (general) Cauchy integral formula implies the following extension of Theorem 5.3 in and Theorem 1 in: for each proper subsector S _θ0, there exist positive constants M, δ depending only on θ₀, such that (δⁿ/n!)∥z ⁿAⁿT(z)x∥ ≤ M ∥x∥ for all n ∈ ℕ, z ∈ S_θ0, and x ∈ D^∞ (A). It follows in particular that the vectors T(z)x (with z ∈ S_θ and x ∈ D^∞ (A)) are analytic vectors for A (hence A has a dense set of analytic vectors).