Spectral theorem for bounded normal operators

Remarks

One proof starts from the continuous functional calculus $f \mapsto f(N)$ on $C(\sigma(N))$. For fixed $x, y$, the map $f \mapsto \ip{f(N) x}{y}$ is a bounded linear functional on $C(\sigma(N))$, so by the Riesz–Markov theorem it is integration against a complex measure $E_{x,y}$. One then defines $E(B)$ by $\ip{E(B) x}{y} = E_{x,y}(B)$ and checks that it is a projection-valued measure. The projections $E(B)$ commute with every bounded operator that commutes with $N$. An equivalent form holds when $H$ is separable. Then there are a $\sigma$-finite measure space $(\Omega, \mu)$, a bounded measurable function $\varphi$ on $\Omega$ and a unitary map $U \colon H \to L^2(\Omega, \mu)$ with $U N U^{-1} f = \varphi f$ for all $f \in L^2(\Omega, \mu)$. Example: on $L^2([0,1])$, the operator $(Nf)(t) = t f(t)$ is self-adjoint with spectrum $[0,1]$, and $E(B)$ is multiplication by the indicator function of $B$. In finite dimension, $E(B)$ is the orthogonal projection onto the sum of the eigenspaces for the eigenvalues in $B$, and the theorem is the spectral theorem for normal matrices. Normality is necessary: the right shift on $\ell^2$ is not normal, and it is not of this form.

Going deeper

For a student

The theorem makes it possible to define $f(N) = \int f(\lambda) \, dE(\lambda)$ for every bounded Borel function $f$ on $\sigma(N)$, the Borel functional calculus. With $f$ an indicator function, this gives the spectral projections of $N$, which split $H$ into pieces where $N$ acts with spectrum in a given set. The self-adjoint case covers the observables of quantum mechanics with bounded values; the unitary case gives the spectral analysis of unitary operators, used in ergodic theory. For unbounded self-adjoint operators, a similar theorem holds with a projection-valued measure on $\R$.