Direct integrals and general multiplicity theory

Assume AC. The standard measurable direct-integral model for a bounded normal operator on a separable Hilbert space, and the classification of such operators by the scalar measure class together with the almost-everywhere multiplicity function, are proved on this page: the model with its measurable field of fibers of dimension $m(z)$ and the identification with the orthogonal sum of cyclic $L^2$-summands are Spectral multiplicity function in the separable case; the invariance of the scalar measure class and of the fiber dimension under unitary intertwiners is Unitary intertwiners preserve direct-integral fiber dimension; and the classification statement is Unitary equivalence classified by measure class and multiplicity. In that theorem the multiplicity is the almost-everywhere dimension of the direct-integral fiber, not the dimension of the eigenspace $\ker (T-zI)$. The two notions can differ drastically: A normal operator need not have any eigenvectors exhibits a normal operator with spectrum $[0,1]$ but no nonzero eigenspace at any spectral point.

General measurable fields of Hilbert spaces beyond the standard countable fibers used here, and nonseparable multiplicity theory, are orientation only: they are not constructed, not stated as results, and are not suppliers for any item on this page or its consumers. The separable statements above are self-contained in the sense that every supplier they use is either proved earlier in the library or earlier on this page.