22.8. GENERAL THEORY OF CONTINUOUS SEMIGROUPS 605

≤∫

∞

0

∥∥∥e−λ tS (t)∥∥∥∥∥∥∥S (h)x− x

h−Ax

∥∥∥∥dt

Then, passing to a limit as h→ 0+, this integrand converges uniformly to 0 so for allλ > α,

limh→0

R(λ )

(S (h)x− x

h

)= R(λ )Ax (22.32)

Also, S (h) commutes with R(λ ) . This follows from approximating with Riemann sumsand taking a limit. Thus also

limh→0

R(λ )

(S (h)x− x

h

)= lim

h→0

(S (h)R(λ )x−R(λ )x

h

)= AR(λ )x

so we have for x ∈ D(A) ,R(λ )Ax = AR(λ )x. However, this implies

R(λ )(λ I−A)x = (λ I−A)R(λ )x = x

from Claim 1. Thus R(λ ) is a left inverse of (λ I−A). Since R(λ ) = (λ I−A)−1 , thisshows the estimate 22.30 from 22.31. This proves Claim 3.

Why is A a closed operator? Suppose xn → x where xn ∈ D(A) and that Axn → ξ . Ineed to show that this implies that x ∈ D(A) and that Ax = ξ . Thus xn→ x and for λ > α,

(λ I−A)xn → λx− ξ . However, 22.30 shows that (λ I−A)−1 = R(λ ) is continuous andso

xn→ (λ I−A)−1 (λx−ξ ) = x

It follows that x ∈ D(A) . Then doing (λ I−A) to both sides of the equation, λx− ξ =λx−Ax and so Ax = ξ showing that A is a closed operator as claimed. ■

Definition 22.8.6 The linear mapping for λ > α where ∥S (t)∥ ≤ Meαt given by(λ I−A)−1 = R(λ ) is called the resolvent.

The following corollary is also very interesting.

Corollary 22.8.7 Let S (t) be a continuous semigroup and let A be its generator. Thenfor 0 < a < b < ∞ and x ∈D(A) , S (b)x−S (a)x =

∫ ba S (t)Axdt and also for t > 0 you can

take the derivative from the left,

limh→0+

S (t)x−S (t−h)xh

= S (t)Ax

Proof: Letting y∗ ∈ X ′, you can take y∗ inside the integral by approximating withRiemann sums. Thus

y∗(∫ b

aS (t)Axdt

)=∫ b

ay∗(

S (t) limh→0

S (h)x− xh

)dt

The difference quotients are bounded because they converge to Ax. Therefore, from thedominated convergence theorem and using the semigroup property,

y∗(∫ b

aS (t)Axdt

)= lim

h→0

∫ b

ay∗(

S (t)S (h)x− x

h

)dt