Exercise 13, § 7

Let E be a complex Hilbert space. Let N be a mapping from $\mathscr{L}(E)$ into $\mathfrak{P}(\mathbf{C})$ such that

a) $N(u)$ is compact for every $u\in \mathscr{L}(E)$ ;

b) $N(\alpha u+\beta 1_E) =\alpha N(u) +\beta$ for every $u\in \mathscr{L}(E)$ and every $(\alpha , \beta )\in \mathbf{C}^2$;

c) $N(u)\subset \{z\in \mathbf{C}|\mathscr{R}(z)\geqslant 0\}$ if and only if $u+u^*$ is positive.

Then $N(u) =\iota (u)$ for every $u\in \mathscr{L}(E)$. (Show that $u\mapsto \iota (u)$ has the indicated properties; let $N_1$ and $N_2$ be mappings satisfying them; supposing that $N_1(u)$ is not contained in $N_2(u)$, find $\alpha$ and $\beta$ in $\mathbf{C}$ such that $N_2(\alpha u+\beta )$ is contained in the half-plane $\mathscr{R}(z)\geqslant 0$, but $N_1(\alpha u+\beta )$ is not, and conclude to a contradiction using c).)