1708 CHAPTER 54. FUNCTIONAL ANALYSIS APPLICATIONS

By assumption |arg(t)| ≤ r <(

π

2 −φ)

and so

arg(w)+ arg(t)≥ (π−φ)− r =π

2+(

π

2−φ

)− r ≡ π

2+δ (r) , δ (r)> 0.

ei(argw+arg t) = cos(argw+ arg t)+ isin(argw+ arg t) , so cos(argw+ arg t)< 0

It follows the integral dominated by an expression of the form

eat 12π

∫∞

1/|t|exp(−c(r) |t|y) M

ydy = eat 1

2π

∫∞

1exp(−c(r)x)

M |t|x

1|t|

dx

= eat 12π

∫∞

1exp(−c(r)x)

Mx

dx

where c(r)< 0 independent of |arg(t)| ≤ r. A similar estimate holds for the integral on thebottom segment. Thus for |arg(t)| ≤ r, ∥S (t)∥ is bounded by Meat for some constant M. Inparticular, ∥S (t)∥e−at is bounded for t ∈ [0,∞).

Now let x ∈ D(A) . From 54.3.6,

eλ t

λ(λ −A)−1 Ax+

eλ t

λx = eλ t (λ I−A)−1 x (54.3.14)

On the circular part of the contour, λ = a+ 1|t|e

iθ . Consider the first term on the left in theabove equation. The contour integral is of the form

∫π−φ

φ−π

eateei(θ+arg(t)) 1a+ 1

|t|eiθ

((a+

1|t|

eiθ)

I−A)−1

Axi|t|

eiθ dθ

which is dominated by

e∣∣eat ∣∣∫ π−φ

φ−π

1∣∣∣a+ 1|t|e

iθ∣∣∣ M∣∣∣ 1|t|e

iθ∣∣∣ ∥Ax∥ 1

|t|≤ eatM̂ ∥Ax∥

∫π−φ

φ−π

|t||a |t|+ eiθ |

dθ

≤ eatM̂ ∥Ax∥∫

π−φ

φ−π

|t|1−|a| |t|

dθ

which converges to 0 as t→ 0. On the other part of the contour, λ = yw+a where arg(w) =π−φ ,y > 1/ |t|.

eat

2πi

∫∞

1/|t|eywt 1

yw+a((yw+a) I−A)−1 wdy

As above, arg(w)+arg(t)> π

2 +δ (r) ,δ (r)> 0 for |arg t| ≤ r <(

π

2 −φ). Thus, as above,

this integral is dominated by

eat

2π

∫∞

1/|t|e−y|t|c(r) 1

|yw+a|M|y|

dy =eat

2π

∫∞

1e−uc(r) |t|

|uw+a |t||M|u|

du