Spectral Theorem

Let $H$ be a Hilbert Space, $B(H)$ the set of Bounded linear operators from $H$ to itself, and $\sigma(T)$ the Spectrum of $T$. Then if $T\in B(H)$ and $T$ is normal, there exists a unique resolution of the identity $E$ on the Borel subsets of $\sigma(T)$ which satisfies

$$T=\int_{\sigma(T)} \lambda\,dE(\lambda).$$

Furthermore, every projection $E(\omega)$ Commutes with every $S\in B(H)$ which Commutes with $T$.

References

Rudin, W. Theorem 12.23 in Functional Analysis, 2nd ed. New York: McGraw-Hill, 1991.