Editing Self-adjoint operator (section)

=== Direct integrals ===
The spectral multiplicity theorem can be reformulated using the language of [[direct integral]]s of Hilbert spaces:

{{math theorem|<ref>{{harvnb|Hall|2013}} Theorems 7.19 and 10.9</ref> Any self-adjoint operator on a separable Hilbert space is unitarily equivalent to multiplication by the function λ ↦ λ on
<math display="block">\int_\mathbf{R}^\oplus H_\lambda\, d \mu(\lambda).</math>}}

Unlike the multiplication-operator version of the spectral theorem, the direct-integral version is unique in the sense that the measure equivalence class of ''μ'' (or equivalently its sets of measure 0) is uniquely determined and the measurable function <math>\lambda\mapsto\mathrm{dim}(H_{\lambda})</math> is determined almost everywhere with respect to ''μ''.<ref>{{harvnb|Hall|2013}} Proposition 7.22</ref> The function <math>\lambda \mapsto \operatorname{dim}\left(H_\lambda\right)</math> is the '''spectral multiplicity function''' of the operator.

We may now state the classification result for self-adjoint operators: Two self-adjoint operators are unitarily equivalent if and only if (1) their spectra agree as sets, (2) the measures appearing in their direct-integral representations have the same sets of measure zero, and (3) their spectral multiplicity functions agree almost everywhere with respect to the measure in the direct integral.<ref>{{harvnb|Hall|2013}} Proposition 7.24</ref>