Exercise 7, § 4

Let E be a complex Banach space and $u$ a partial operator on E. One says that $u$ is accretive if, for every $x\in$ dom($u$), there exists a continuous linear form $\ell \in E'$ such that

(i) $\mathscr{R}(\ell (u(x)))\geqslant 0$;

(ii) one has $\|\ell \|=\|x\|$ and $\ell (x) =\|x\|^2$.

One says that a continuous semi-group $s$ on E is contracting if $\|s(t)\|\leqslant 1$ for every $t\in \mathbf{R}_+$.

a) The partial operator $u$ is the infinitesimal generator of a contracting continuous semi-group on E if and only if $u$ is accretive and if there exists $\lambda >0$ such that the partial operator $\lambda 1_E+u$ is surjective. (To prove that the condition is necessary, let $s$ be the contracting semi-group whose infinitesimal generator is $u$; for $x\in$ dom($u$), consider the function defined by $t\mapsto \ell (s(t)x)$; it is differentiable at 0, and its derivative at 0 is $-\mathscr{R}(\ell (u(x))).$)

b) Suppose that $u$ is closed and that $u$ and $^tu$ are accretive. Then $u$ is the infinitesimal generator of a contracting continuous semi-group on E.