596 CHAPTER 19. HILBERT SPACES

Thus−αI+A generates e−αtS (t) and it follows from the first part that−αI+A∗ generatesthe semigroup e−αtS∗ (t) . Thus

−αx+A∗x = limh→0+

e−αhS∗ (h)x− xh

= limh→0+

(e−αh S∗ (h)x− x

h+

e−αh−1h

x)

= −αx+ limh→0+

S∗ (h)x− xh

showing that A∗ generates S∗ (t) . It follows from Proposition 19.14.5 that A∗ is closed anddensely defined. It is obvious S∗ (t) is a semigroup. Why is it continuous? This also followsfrom the first part of the argument which establishes that

t→ e−αtS∗ (t)x

is continuous. This proves the theorem.