Corollary

Let $u$ be a self-adjoint operator on E.

a) The residual spectrum of $u$ is empty;

b) For all $\lambda \not =\mu$ in $\mathbf{C}$, the eigenspaces of $u$ corresponding to $\lambda$ and $\mu$ are orthogonal.

Let us prove a). Let $\lambda$ belong to the spectrum of $u$; it is a real number (Prop. 17). One has Ker($\lambda 1_E-u$) $=$ Im($\lambda 1_E-u$)$^{\circ}$ (Prop. 7 of IV, p. 236), therefore $\lambda$ is not an eigenvalue of $u$ if the partial operator $\lambda 1_E-u$ has dense image. By definition this implies that the residual spectrum of $u$ is empty.

Let us prove b). By the proposition, we may suppose that $\lambda$ and $\mu$ are real. Let $x\in$ dom($u$) be such that $u(x) =\lambda x$ and $y\in$ dom($u$) be such that $u(y) =\mu y$. We then have

whence $\langle x|y\rangle = 0$.