574 CHAPTER 19. HILBERT SPACES

=∫

∞

0

((εI +A)e−(εI+A)t − (εI +A)e−(δ I+A)t

)tα−1dt

+∫

∞

0

((εI +A)e−(δ I+A)t − (δ I +A)e−(δ I+A)t

)tα−1dt

= Pδ ,ε +Qδ ,ε .

Then Pδ ,ε = ∫∞

0(εI +A)

(e−(εI+A)t − e−(δ I+A)t

)tα−1dt

∥∥Pδ ,ε

∥∥≤ ∫ ∞

0∥(εI +A)∥

∥∥∥(e−(εI+A)t − e−(δ I+A)t)∥∥∥ tα−1dt

We need to estimate the difference of those semigroups. Call the first, e−(εI+A)t ≡ y andthe second x. Then by definition,

y′+(εI +A)y = 0,y(0) = x0

x′+(δ I +A)x = 0,x(0) = x0

Then ŷ(t)≡ eεty(t) , x̂(t)≡ eδ tx(t)

ŷ′− ε ŷ(t)+(εI +A) ŷ = 0, ŷ(0) = x0

x̂′−δ x̂(t)+(δ I +A) x̂ = 0, x̂(0) = x0

Thus

ŷ′+Aŷ = 0, ŷ(0) = x0

x̂′+Ax̂ = 0, x̂(0) = x0

By uniqueness, x̂ = ŷ. Thus ∣∣∣eεty(t)− eδ tx(t)∣∣∣= 0

Soeδ t∣∣∣e(ε−δ )ty(t)− x(t)

∣∣∣= 0

So, since x0 was arbitrary,

e(ε−δ )te−(εI+A)t − e−(δ I+A)t = 0

Then ∥∥Pδ ,ε

∥∥≤C (∥A∥)∫

∞

0

∥∥∥(e−(εI+A)t − e(ε−δ )te−(εI+A)t)∥∥∥ tα−1dt

≤C (∥A∥)∫

∞

0

∣∣∣1− e(ε−δ )t∣∣∣ tα−1dt =C (∥A∥)

∫∞

0

(1− e−(δ−ε)t

)tα−1dt

Now we need an estimate. Suppose 1 > λ > 0. For t ≥ 0, is

1− e−λ t ≤ λe−λ t?