Corollary (Spectral Theorem)

Let E be a complex Hilbert space. Let $u\in \mathscr{L}(E)$ be a normal endomorphism of E. There exist a locally compact topological space X, a positive measure $\mu$ on X, an isometric isomorphism $\theta$ of $L^2(X, \mu)$ onto E, and a continuous bounded function $g$ on X such that $u=\theta \circ m_g\circ \theta^{-1}$.

If $u$ admits a cyclic vector $x$, one may take X = Sp($u$) and for $g$ the identity function of X.

It is enough to apply theorem 1 (resp. the corollary of proposition 8) to the involutive subalgebra A of $\mathscr{L}(E)$ generated by $u$, and to put $g=\pi (u)$.

This statement reduces every question concerning a single normal endomorphism of a Hilbert space to a similar question for a multiplication operator, which often simplifies its study (cf. for example exercise 19 of IV, p. 325).