Theorem 2.3.2 (Spectral Theorem I, [The Spectral Theorem, Con85]).label Let $H$ be a Hilbert space and $T$ be a normal operator, then there exists a unique spectral measure $P: \cb_{\complex} \to L(H; H)$ such that:

1. (1)

   $T = \int z P(dz)$.

2. (2)

   The mapping $f \mapsto \int f(z) P(dz)$ is a $*$-homomorphism.