19.14. GENERAL THEORY OF CONTINUOUS SEMIGROUPS 577

Theorem 19.14.3 For M described in Lemma 19.14.2, there exists α such that

∥S (t)∥ ≤Meαt , t ≥ 0

In fact, α can be chosen such that M1/T = eα .

Proof: Let t be arbitrary. Then t = mT + r (t) where 0 ≤ r (t) < T . Then by the semi-group property

∥S (t)∥= ∥S (mT + r (t))∥= ∥S (r (t))S (T )m∥ ≤Mm+1

Now mT ≤ t ≤ mT + r (t)≤ (m+1)T and so m≤ tT ≤ m+1. Therefore,

∥S (t)∥ ≤M(t/T )+1 = M(

M1/T)t.

Let M1/T ≡ eα and then ∥S (t)∥ ≤Meαt

Definition 19.14.4 Let S (t) be a continuous semigroup as described above. It is called acontraction semigroup if for all t ≥ 0

∥S (t)∥ ≤ 1.

It is called a bounded semigroup if there exists M such that for all t ≥ 0,

∥S (t)∥ ≤M

Note that for S (t) an arbitrary continuous semigroup satisfying ∥S (t)∥ ≤Meαt , It fol-lows that the semigroup, T (t) = e−αtS (t) is a bounded semigroup which satisfies ∥T (t)∥≤M.

The next proposition has to do with taking a Laplace transform of a semigroup.

Proposition 19.14.5 Given a continuous semigroup S (t) , its generator A exists and is aclosed densely defined operator. Furthermore, for

∥S (t)∥ ≤Meαt

and λ > α, λ I−A is one to one and onto from D(A) to X. Also (λ I−A)−1 maps X ontoD(A) and is in L (X ,X). Also for these values of λ > α,

(λ I−A)−1 x =∫

∞

0e−λ tS (t)xdt.

For λ > α, the following estimate holds.∥∥∥(λ I−A)−1∥∥∥≤ M|λ −α|

(19.14.79)