Proposition 12

Tag 081I, permanent. Cite it as [Bourbaki, Tag 081I].

Topological Vector Spaces, chapter V, § 4. Some classes of operators in hilbertian spaces

— Let u be a hermitian (resp. positive) element of $\mathscr{L}(E)$ and let v be a continuous linear mapping from E into a hilbertian space F. Then $vuv^*$ is a hermitian (resp. positive) element of $\mathscr{L}(F)$ .

For, we have $(vuv^*)^* = v^{**}u^*v^* = vuv^*$ . On the other hand, if $u \geq 0$ , we have

\langle y|vuv^*(y) \rangle = \langle v^*(y)|u(v^*(y)) \rangle \geq 0

for all $y \in F$ , hence $vuv^* \geq 0$ .

Prop. 12 shows, in particular, that $vv^*$ is positive for all $v \in \mathscr{L}(E; F)$ . Since, in particular, an orthoprojector $p$ satisfies $p = p^2 = pp^*$ , it is positive.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.