Definition 6

— Let E be a hilbertian space and $u \in \mathscr{L}(E)$. We say that u is positive, and write $u \geq 0$, if u is hermitian and if $\langle x|u(x) \rangle \geq 0$ for all $x \in E$.

When K is equal to C, the relation

implies that u is hermitian (V, p. 2, Remark), hence positive.

Let $\mathscr{L}_+(E)$ denote the set of all positive elements of $\mathscr{L}(E)$; this is a proper pointed convex cone in the real vector space $\mathscr{L}(E)_{[\mathbf{R}]}$ underlying $\mathscr{L}(E)$. In order that u is positive, it is necessary and sufficient that the sesquilinear form $\Phi_u$ on $E \times E$ associated with u is positive hermitian. Given u and v in $\mathscr{L}(E)$, the relation $u - v \geq 0$ can also be written as $u \geq v$ or $v \leq u$; this is an order relation on $\mathscr{L}(E)_{[\mathbf{R}]}$ compatible with its real vector space structure.