Exercise 5, § 4

Let E be a complex Banach space. A continuous semigroup on E is a mapping $s:\mathbf{R}_+\rightarrow \mathscr{L}(E)$ such that

(i) $s(0) = 1_E$;

(ii) The mapping of $\mathbf{R}_+\times E\rightarrow E$ defined by $(t, x)\mapsto s(t)x$ is continuous;

(iii) For all $t_1$ and $t_2$ in $\mathbf{R}_+$, one has $s(t_1+t_2) =s(t_1)s(t_2)$.

We consider a continuous semigroup $s$ on E.

a) There exist real numbers $C\geqslant 0$ and $M\geqslant 0$ such that

for every $t\in \mathbf{R}_+$. (Let $a >0$; now prove that $\|s(t)\|$ is bounded for $t\in [0, a].$) Such numbers are fixed in the rest of the exercise.

b) Let D be the set of $x\in E$ such that the mapping $\psi_x$ of $\mathbf{R}_+$ into E defined by $t\mapsto s(t)x$ is differentiable at 0. The space D is dense in E; the mapping $u$ of D into E defined by $u(x) =-\psi '_x(0)$ is a closed partial operator on E, which is called the infinitesimal generator of the semigroup $s$. c) Let $\lambda$ be a complex number such that $\mathscr{R}(\lambda )<-M$. The formula

defines an endomorphism of E.

d) The image of $v_{\lambda}$ is contained in dom($u$) and one has $u\circ v_{\lambda}=-1_E-\lambda v_{\lambda}$. e) One has $u\circ v_{\lambda}=v_{\lambda}\circ u$.

f) The complex number $\lambda$ belongs to the resolvent set of $u$ and one has $v_{\lambda}=-R(u, \lambda )$.

g) One has

for every $n\in \mathbf{N}$.