614 CHAPTER 22. HILBERT SPACES

With this preparation, here is an interesting result about the adjoint of the generator ofa continuous bounded semigroup. I found this in Balakrishnan [5].

Theorem 22.8.14 Suppose A is a densely defined closed operator which generatesa continuous semigroup, S (t) . Then A∗ is also a closed densely defined operator whichgenerates S∗ (t) and S∗ (t) is also a continuous semigroup.

Proof: First suppose S (t) is also a bounded semigroup, ∥S (t)∥ ≤ M. From Lemma22.8.13 A∗ is closed and densely defined. It follows from the Hille Yosida theorem, Theo-rem 22.8.8 that ∣∣∣((λ I−A)−1

)n∣∣∣≤ Mλ

n

From Lemma 22.8.13 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 22.8.8 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 proof ofTheorem 22.8.8 and Lemma 22.8.13, it follows that for x ∈ D(A∗) and a suitable sequence{λ n} ,

(T (t)x,y) =

 limn→∞

e−λ nt∞

∑k=0

tk(

λ2n (λ nI−A∗)−1

)k

k!x,y



= limn→∞

e−λ nt∞

∑k=0

tk((

λ2n (λ nI−A)−1

)k)∗

k!x,y



= limn→∞

x,e−λ nt

 ∞

∑k=0

tk((

λ2n (λ nI−A)−1

)k)

k!

y

= (x,S (t)y) = (S∗ (t)x,y) .

Therefore, since y is arbitrary, S∗ (t) = T (t) on x ∈ D(A∗) a dense set and this shows thetwo are equal. This proves the proposition in the case where S (t) is also bounded.

Next only assume S (t) is a continuous semigroup. Then by Proposition 22.8.5 thereexists α > 0 such that

∥S (t)∥ ≤Meαt .

Then consider the operator −αI +A and the bounded semigroup e−αtS (t). For x ∈ D(A)

limh→0+

e−αhS (h)x− xh

= limh→0+

(e−αh S (h)x− x

h+

e−αh−1h

x)

= −αx+Ax