54.3. SECTORIAL OPERATORS AND ANALYTIC SEMIGROUPS 1711

=1

2πi

∫γε,φ

eλ tA(λ I−A)−1 dλ

because of Lemma 54.3.6 the Cauchy integral theorem, and approximating γε,φ with closedcontours.

Now from Lemma 54.3.6 one can take A out of the inte-gral and

S′ (t) = A

(1

2πi

∫γε,φ

eλ t (λ I−A)−1 dλ

)= AS (t)

To get the higher derivatives, note S (t) has infinitely many derivatives due to t being acomplex variable. Therefore,

S′′ (t) = limh→0

S′ (t +h)−S′ (t)h

= limh→0

AS (t +h)−S (t)

h

and S(t+h)−S(t)h → AS (t) and so since A is closed, AS (t) ∈ D(A) and the above becomes

A2S (t). Continuing this way yields the claims 1.) and 2.). Note this also implies S (t)x ∈D(A) for each t ∈ S0

0(φ+π/2) which says more than S (t)x ∈H. In practice this has the effectof regularizing the solution to an initial value problem.

Next consider the semigroup property. Let s, t ∈ S00,(φ+π/2). As described above let γ ′

ε,φ

denote the contour shifted slightly to the right. Then

S (t)S (s) =(

12πi

)2 ∫γε,φ

∫γ ′

ε,φ

eλ t (λ I−A)−1 eµs (µI−A)−1 dµdλ (54.3.16)

Using the resolvent identity,

(λ I−A)−1 (µI−A)−1 = (µ−λ )−1((λ I−A)−1− (µI−A)−1

),

then substituting this resolvent identity in 54.3.16, it equals(1

2πi

)2 ∫γε,φ

∫γ ′

ε,φ

eµseλ t((µ−λ )−1

((λ I−A)−1− (µI−A)−1

))dµdλ

= −(

12πi

)2 ∫γε,φ

eλ t∫

γ ′ε,φ

eµs (µ−λ )−1 (µI−A)−1 dµdλ

+

(1

2πi

)2 ∫γε,φ

∫γ ′

ε,φ

eµseλ t (µ−λ )−1 (λ I−A)−1 dµdλ

The order of integration can be interchanged because of the absolute convergence and Fu-bini’s theorem. Then this reduces to

= −(

12πi

)2 ∫γ ′

ε,φ

(µI−A)−1 eµs∫

γε,φ

eλ t (µ−λ )−1 dλdµ

+

(1

2πi

)2 ∫γε,φ

(λ I−A)−1 eλ t∫

γ ′ε,φ

eµs (µ−λ )−1 dµdλ