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10 says that the spectrum of a semigroup generator is always contained in a left half-plane. The number determining the smallest such half-plane is an important characteristic of any linear operator and is now defined explicitly. 12 Definition. With any linear operator A we associate its spectral bound defined by s(A) := sup{Re λ : λ ∈ σ(A)}. 5) and the spectral bound of its generator. 13 Corollary. For a strongly continuous semigroup T (t) erator A, one has −∞ ≤ s(A) ≤ ω0 < +∞. 14 Diagram. To conclude this section, we collect in a diagram the information obtained so far on the relations between a semigroup, its generator, and its resolvent.

For the orbit map ξx : t → T (t)x, the following properties are equivalent. (a) ξx (·) is differentiable on R+ . (b) ξx (·) is right differentiable at t = 0. Proof. We have only to show that (b) implies (a). For h > 0, one has lim h1 T (t + h)x − T (t)x = T (t) lim h1 T (h)x − x h↓0 h↓0 = T (t) ξ˙x (0), and hence ξx (·) is right differentiable on R+ . On the other hand, for −t ≤ h < 0, we write 1 T (t + h)x − T (t)x − T (t)ξ˙x (0) = T (t + h) h 1 h x − T (−h)x − ξ˙x (0) + T (t + h)ξ˙x (0) − T (t)ξ˙x (0).

Assertion (i) is trivial. To prove (ii) take x ∈ D(A). 1) that 1/h T (t + h)x − T (t)x converges to T (t)Ax as h ↓ 0. 4) with AT (t)x = T (t)Ax. The proof of assertion (iii) is included in the following proof of (iv). For x ∈ X and t ≥ 0, one has 1 T (h) h t t T (s)x ds − 0 = = = T (s)x ds 0 1 h 1 h 1 h t 0 t+h T (s)x ds − h t+h T (s)x ds − t t 1 h T (s + h)x ds − 1 h 1 h T (s)x ds 0 t T (s)x ds 0 h T (s)x ds, 0 which converges to T (t)x − x as h ↓ 0. 6) holds. If x ∈ D(A), then the functions s → T (s) (T (h)x−x)/h converge uniformly on [0, t] to the function s → T (s)Ax as h ↓ 0.

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