structural:reformatting
Spectral theorem

There are many theorems known as the spectral theorem, but one in particular has many applications in functional analysis.

Spectral theorem — Let A {\displaystyle A} be a bounded self-adjoint operator on a Hilbert space H {\displaystyle H}. Then there is a measure space (X, Σ, μ) {\displaystyle (X,\Sigma,\mu)} and a real-valued essentially bounded measurable function f {\displaystyle f} on X {\displaystyle X} and a unitary operator U: H → L μ 2 (X) {\displaystyle U:H\to L_{\mu }^{2}(X)} such that U ∗ T U = A {\displaystyle U^{*}TU=A} where T is the multiplication operator: (x) = f (x) φ (x). {\displaystyle (x)=f(x)\varphi (x).} and ‖ T ‖ = ‖ f ‖ ∞ {\displaystyle \|T\|=\|f\|_{\infty }}.

This is the beginning of the vast research area of functional analysis called operator theory; see also the spectral measure.

There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces. The only difference in the conclusion is that now f {\displaystyle f} may be complex-valued.