436 CHAPTER 17. SPECTRAL THEORY OF LINEAR MAPS

and from Lemma 17.4.8 these two sides are dominated by

|eat |π

∫∞

1/|t|e−δ ry|t| 1

|yw+a|My

dy∥Ax∥

Now letting u = |t|y this equals

|eat |π

∫∞

1e−δ ru 1∣∣∣ u

|t|w+a∣∣∣ |t|Mu 1

|t|du∥Ax∥= |e

at |π

∫∞

1e−δ ru |t|

|uw+ |t|a|Mu

du∥Ax∥

Which converges to 0 as t → 0 in the sector |arg t| ≤ r <(

π

2 −φ). Thus from 17.26, for

t ∈ Sr

S (t)x = ε (t)+1

2πi

∫γ

eλ t

λxdλ , lim

t→0ε (t) = 0 (17.27)

Now approximate γ with a closed contour having a large circular arc of radius R such thatthe resulting bounded contour γR has 0 on its inside and∥∥∥∥∥∥∥∥∥

12πi

∫γ

eλ t

λxdλ −

=x︷ ︸︸ ︷1

2πi

∫γR

eλ t

λxdλ

∥∥∥∥∥∥∥∥∥< η (R)

where limR→∞ η (R) = 0. By the Cauchy integral formula, 12πi∫

γReλ t

λxdλ = x and so, from

this, the above, and 17.27,

∥S (t)x− x∥ ≤

∥∥∥∥∥ε (t)+1

2πi

∫γ

eλ t

λxdλ − x

∥∥∥∥∥≤ ∥ε (t)∥+η (R)

Let R→ ∞ and then it follows limt→0 ∥S (t)x− x∥ = limt→0 ∥ε (t)∥ = 0. By the first part,∥S (t)∥ is bounded for small t in Sr so it follows that, since D(A) is dense, then for anyx ∈H, It follows that limt→0,t∈Sr S (t)x = x where t is in the sector Sr given by |arg t| ≤ r <(

π

2 −φ).

Now for |arg t| ≤ r <(

π

2 −φ), AS (t) = 1

2πi∫

γε,φeλ tA(λ I−A)−1 dλ . From 17.13 this

is1

2πi

∫γ

eλ t(−I +λ (λ I−A)−1

)dλ

On the circle, λ = a+ 1|t|e

iθ and as above, this is

∫π−φ

φ−π

eateei(θ+arg(t))

(−I +

(a+

1|t|

eiθ)((

a+1|t|

eiθ)

I−A)−1

)i|t|

eiθ dθ

and by the estimates and letting M > 1, this is dominated by

e∣∣eat ∣∣∫ π−φ

φ−π

1+M

∣∣∣a+ 1|t|e

iθ∣∣∣

1/ |t|

 1|t|

dθ ≤ e∣∣eat ∣∣M ∫

π−φ

φ−π

(1+∣∣∣a |t|+ eiθ

∣∣∣) 1|t|

dθ