Semigroups of Operators in Banach Space

We begin with a definition Definition 1. A one-parameter family T (t) for 0 ≤ t < ∞ of bounded linear operators on a Banach space X is a C 0 (or strongly continuous) Semigroup on X if 1. T (0) = I (the identity on X). 2. T (t + s) = T (t)T (s) (semigroup property) 3. lim t↓0 T (t)v = v for all v ∈ X (This the Strong Continuity at t = 0, i.e., continuous at t = 0 in the strong operator topology). Theorem 1. If T (t) is a C 0 semigroup, then there exists ω ≥ 0 and M ≥ 1 such that T (t) ≤ M e ωt ∀ t ≥ 0. (1) Proof: 1. First we show that T (t) is bounded on 0 ≤ t ≤ η for some η > 0. If not then there exist {t n } such that t n ↓ 0 and T (t n) ≥ n. From the principle of uniform boundedness there must exist a v ∈ X such that T (t n)v is unbounded. But this contradicts the strong continuity at t = 0. Thus there exists η and M such that T(t) ≤ M for 0 ≤ t ≤ η. Also since T (0) = 1 we must have M ≥ 1. 2. Let ω = ln(M) η. Given any t ≥ 0 there exists n ∈ Z and δ with 0 ≤ δ ≤ η so that t = nη + δ. Then by the semigroup property we have T(t) = T (δ)T (η) n ≤ M n+1. 3. Now t = nη + δ implies n = (t − δ) η ≤ t η and since M ≥ 1 we have T(t) ≤ M M n ≤ M M t/η. Now ω = ln(M)/η implies ln(M) = ηω which implies M = e ωη. Thus we have T(t) ≤ M M t/η = M e ωη t/η = M e ωt .