582 CHAPTER 19. HILBERT SPACES

Lemma 19.14.9 There is a bounded linear operator given for λ > 0 by

−λ I +λ2 (λ I−A)−1 = λA(λ I−A)−1 ≡ Aλ

On D(A) ,Aλ = λ (λ I−A)−1 A.

For x ∈ D(A) , ∥∥∥λ (λ I−A)−1 x− x∥∥∥

=∥∥∥(λ I−A)−1 (λx− (λ I−A)x)

∥∥∥=

∥∥∥(λ I−A)−1 Ax∥∥∥≤ M

λ∥Ax∥ (19.14.85)

which converges to 0 as λ → ∞.Now Lλ x→ x on a dense subset of X ,Lλ ≡ λ (λ I−A)−1. Also, from the hypothesis,

∥Lλ∥ ≤M. Say x is arbitrary. Then does Lλ x→ x? Let x̂ ∈ D(A) and ∥x− x̂∥< ε. Then

∥Lλ x− x∥ ≤ ∥Lλ x−Lλ x̂∥+∥Lλ x̂− x̂∥+∥x̂− x∥< Mε + ε + ε

whenever λ is large enough and so for all x ∈ X ,

limλ→∞

λ (λ I−A)−1 x = x

In particular, this holds whenever x is replaced with Ax for some x ∈ D(A) . Thus if x ∈D(A) ,

limλ→∞

∥Aλ x−Ax∥= limλ→∞

∥∥∥λ (λ I−A)−1 Ax−Ax∥∥∥= 0 (19.14.86)

This is summarized in the following lemma.

Lemma 19.14.10 For all x ∈ D(A) , limλ→∞ ∥Aλ x−Ax∥= 0.

Now from Corollary 19.12.5, there exists an approximate continuous semigroup Sλ (t)generated by Aλ which is the solution to

S′λ(t) = Aλ Sλ (t) ,Sλ (0) = I (19.14.87)

In terms of power series,

Sλ (t)≡ e−λ t∞

∑k=0

tk(

λ2 (λ I−A)−1

)k

k!= et(−λ I+λ

2(λ I−A)−1) (19.14.88)

Thus, by assumption and triangle inequality,

∥Sλ (t)∥ ≤ e−λ t∞

∑k=0

tk

k!λ

2k M

λk = e−λ tMeλ t = M (19.14.89)