Definition 5

Let E be a hilbertian space and let $u \in \mathscr{L}(E)$. We say that u is hermitian if $u^* = u$.

Let $\mathscr{H}(E)$ denote the set of all hermitian elements of $\mathscr{L}(E)$; this is a vector subspace of the vector space $\mathscr{L}(E)_{[\mathbf{R}]}$ over $\mathbf{R}$ which is deduced from $\mathscr{L}(E)$ by restricting the scalars.

To each $u \in \mathscr{L}(E)$, we associated (V, p. 16, cor. 2) a sesquilinear form $\Phi_u : (x, y) \mapsto \langle x|u(y)\rangle$ on $E \times E$. We have

consequently, $u$ is hermitian if and only if the form $\Phi_u$ is hermitian. When K is $\mathbf{C}$, it is enough to assume that $\Phi_u(x, x) = \langle x|u(x)\rangle$ is real for all $x \in E$ (V, p. 2, Remark).

Let $u \in \mathscr{L}(E)$. We have seen (V, p. 16, cor. 2) that the norm of $u$ can be calculated by the formula

When $u$ is hermitian, we have the following result :