Chapter 10

Definition 10.1. A semigroup of class (C0) is a one parameter family {S(t) | t ≥ 0} ⊂ B(E) satisfying the conditions:

(i) S(t) ◦ S(s) = S(t+ s), for t, s ≥ 0. (ii) S(0) = I.

(iii) the map t→ S(t)(x) is strongly continuous, for each x ∈ E, i.e˙, lim t→0

S(t)x = x, ∀x ∈ E.