Exercise 6, § 4

Let E be a complex Banach space and $u$ a closed partial operator with dense domain on E. Suppose that there exist real numbers $C\geqslant 0$ and $M\geqslant 0$ such that

(i) The set $]-\infty ,-M[$ is contained in the resolvent set of $u$;

(ii) For every integer $n\in \mathbf{N}$ and every real number $\lambda <-M$, one has

a) For every real number $y <-M$, put

The endomorphisms $u_y$ commute pairwise; for every $x\in$ dom($u$), one has $u_y(x)\rightarrow u(x)$ when $y\rightarrow -\infty$.

b) The mapping from $\mathbf{R}_+$ into $\mathscr{L}(E)$ defined by $s_y(t) =e^{-tu_y}$ is a continuous semi-group on E such that $\|s_y(t)\|\leqslant Ce^{Mt}$ for every $t\in \mathbf{R}_+$.

c) For all $y_1,y_2$ and $t$ in $\mathbf{R}^*_+$, and every $x$ in dom($u$), one has

d) Let $t\in \mathbf{R}_+$. For every $x$ in E, the function defined on $\mathbf{R}^*_+$ by $y\mapsto s_y(t)x$ converges when $y\rightarrow -\infty$ to an element $s(t)x$ of E. The mapping $s(t)$ is linear and continuous of norm $\leqslant Ce^{Mt}$.

e) The mapping $t\mapsto s(t)$ is a continuous semi-group on E.

f) The partial operator $u$ is the infinitesimal generator of the continuous semi-group $s$ ("Hille–Yosida theorem"). (First verify that the infinitesimal generator of $s$ is an extension of $u.$)

g) Let $\widetilde{s}$ be a continuous semi-group of which $u$ is the infinitesimal generator. One has $\widetilde{s}=s$.