434 CHAPTER 17. SPECTRAL THEORY OF LINEAR MAPS

Now arg(tw) = arg(t)+arg(w)∈ (π−φ)+[−r,r]∈(

π

2 ,3π

2

)on the top line and arg(tw)∈

3π

2 − φ + [−r,r] ∈(

π

2 ,3π

2

)on the bottom line. Either way, there exists δ r > 0 such that

cos(arg(tw)) ≤ −δ r. This shows 17.20 and implies the existence of the integral definingS (t).

For the last claim, consider the part of γ,γ ′ contained in TR ≡ {z : |Im(z)| ≤ R}. Let hRbe the horizontal parts of the oriented closed contours shown below and let γR and γ ′R bethose parts of γ,γ ′ in TR.

Then from the Cauchy integral theorem,∣∣∣∣−∫γR

eλ t (λ I−A)−1 dλ +∫

γ ′R

eλ t (λ I−A)−1 dλ +∫

hR

eλ t (λ I−A)−1 dλ

∣∣∣∣= 0

From the resolvent estimate as R→ ∞ the last integral in the above converges to 0 and so,passing to a limit one obtains

∣∣∣∫γ ′ eλ t (λ I−A)−1 dλ −

∫γ

eλ t (λ I−A)−1 dλ

∣∣∣= 0. ■

Lemma 17.4.9 Let f (λ ) ,A f (y) be bounded, | f (λ )| , |A f (λ )|< M and continuous onγ∗d,φ and have values in D(A). Then A

∫γd,φ

eλ t f (λ )dλ =∫

γd,φeλ tA f (λ )dλ if t ∈ Sr.

Proof: This follows from the above estimate and noting that for |λ | large the integrandis dominated by an exponential with negative exponent. Therefore, one can approximate theintegrals over those straight segments with integrals over segments of finite length. Thenusing the obvious conclusion for Riemann sums followed by passing to the limit using A isclosed, the desired equation follows. ■

Because of this lemma, I will move A into and out of the integrals which occur in whatfollows.

Next is consideration of the above definition along with estimates.

Lemma 17.4.10 The above of S(t) of Definition 17.4.7

S (t)≡ 12πi

∫γ

eλ t (λ I−A)−1 dλ

is well defined for t ∈ Sr. Also there is a constant Mr such that

∥S (t)∥ ≤Mr∣∣eat ∣∣= MreaRe(t) (17.22)

for every t ∈ Sr such that |arg t| ≤ r <(

π

2 −φ). If Sr is the sector just described, t such that

|arg t| ≤ r <(

π

2 −φ), then for any x ∈ H,

limt→0,t∈Sr

S (t)x = x (17.23)

Also, for |arg t| ≤ r <(

π

2 −φ)∥AS (t)∥ ≤Mr

∣∣eat ∣∣ 1|t|

+Nr∣∣eat ∣∣ |a| (17.24)