590 CHAPTER 19. HILBERT SPACES

Now it follows∣∣∣((λ I−A)x,

((λ I−A)−1

)∗y)∣∣∣≤ |y| |x| for any x ∈ D(A) and so(

(λ I−A)−1)∗

y ∈ D(A∗)

Hence(x,y) =

(x,(λ I−A∗)

((λ I−A)−1

)∗y).

Since x ∈ D(A) is arbitrary and D(A) is dense, it follows

(λ I−A∗)((λ I−A)−1

)∗= I (19.14.98)

From 19.14.97 and 19.14.98 it follows

(λ I−A∗)−1 =((λ I−A)−1

)∗and (λ I−A∗) is one to one and onto with continuous inverse. Finally, from the above,(

(λ I−A∗)−1)n

=((

(λ I−A)−1)∗)n

=((

(λ I−A)−1)n)∗

.

This proves the lemma.With this preparation, here is an interesting result about the adjoint of the generator of

a continuous bounded semigroup. I found this in Balakrishnan [12].

Theorem 19.14.15 Suppose A is a densely defined closed operator which generates a con-tinuous semigroup, S (t) . Then A∗ is also a closed densely defined operator which generatesS∗ (t) and S∗ (t) is also a continuous semigroup.

Proof: First suppose S (t) is also a bounded semigroup, ||S (t)|| ≤ M. From Lemma19.14.14 A∗ is closed and densely defined. It follows from the Hille Yosida theorem, The-orem 19.14.8 that ∣∣∣((λ I−A)−1

)n∣∣∣≤ Mλ

n

From Lemma 19.14.14 and the fact the adjoint of a bounded linear operator preserves thenorm,

Mλ

n ≥∣∣∣(((λ I−A)−1

)n)∗∣∣∣= ∣∣∣(((λ I−A)−1)∗)n∣∣∣

=∣∣∣((λ I−A∗)−1

)n∣∣∣and so by Theorem 19.14.8 again it follows A∗ generates a continuous semigroup, T (t)which satisfies ||T (t)|| ≤M. I need to identify T (t) with S∗ (t). However, from the proofof Theorem 19.14.8 and Lemma 19.14.14, it follows that for x ∈ D(A∗) and a suitablesequence {λ n} ,

(T (t)x,y) =

 limn→∞

e−λ nt∞

∑k=0

tk(

λ2n (λ nI−A∗)−1

)k

k!x,y

