19.14. GENERAL THEORY OF CONTINUOUS SEMIGROUPS 581

Then there exists a continuous semigroup S (t) which has A as its generator and satisfies∥S (t)∥ ≤M and A is closed. In fact letting

Sλ (t)≡ exp(−λ +λ

2 (λ I−A)−1)

it follows limλ→∞ Sλ (t)x = S (t)x uniformly on finite intervals. Conversely, if A is thegenerator of S (t) , a bounded continuous semigroup having ∥S (t)∥≤M, then (λ I−A)−1 ∈L (X ,X) for all λ > 0 and 19.14.83 holds.

Proof: The condition 19.14.83 implies, that∥∥∥(λ I−A)−1∥∥∥≤ M

λ

Consider, for λ > 0, the operator which is defined on D(A) ,

λ (λ I−A)−1 A

On D(A) , this equals−λ I +λ

2 (λ I−A)−1 (19.14.84)

because

(λ I−A)λ (λ I−A)−1 A = λA

(λ I−A)(−λ I +λ

2 (λ I−A)−1)

= −λ (λ I−A)+λ2 = λA

and, by assumption, (λ I−A) is one to one so these are the same. However, the second onein 19.14.84, −λ I +λ

2 (λ I−A)−1 makes sense on all of X . Also(−λ I +λ

2 (λ I−A)−1)(λ I−A) =−λ (λ I−A)+λ

2I = λA

λA(λ I−A)−1 (λ I−A) = λA

so, since (λ I−A) is onto, it follows that on X ,

−λ I +λ2 (λ I−A)−1 = Aλ (λ I−A)−1 ≡ Aλ

Denote this as Aλ to save notation. Thus on D(A) ,

λA(λ I−A)−1 = λ (λ I−A)−1 A = Aλ

although the λ (λ I−A)−1 A only makes sense on D(A). Note that formally

limλ→∞

λ (λ I−A)−1 A = A

This is summarized next.