Editing Self-adjoint operator (section)

== Spectral multiplicity theory ==
The multiplication representation of a self-adjoint operator, though extremely useful, is not a canonical representation. This suggests that it is not easy to extract from this representation a criterion to determine when self-adjoint operators ''A'' and ''B'' are unitarily equivalent. The finest grained representation which we now discuss involves spectral multiplicity. This circle of results is called the ''[[Hans Hahn (mathematician)|Hahn]]–[[Ernst Hellinger|Hellinger]] theory of spectral multiplicity''.

=== Uniform multiplicity ===
We first define ''uniform multiplicity'':

'''Definition'''. A self-adjoint operator ''A'' has uniform multiplicity ''n'' where ''n'' is such that 1 ≤ ''n'' ≤ ''ω'' if and only if ''A'' is unitarily equivalent to the operator M<sub>''f''</sub> of multiplication by the function ''f''(''λ'') = ''λ'' on
: <math>L^2_\mu\left(\mathbf{R}, \mathbf{H}_n\right) = \left\{\psi: \mathbf{R} \to \mathbf{H}_n: \psi \text{ measurable and } \int_{\mathbf{R}} \|\psi(t)\|^2 d\mu(t) < \infty\right\}</math>
where '''H'''<sub>''n''</sub> is a Hilbert space of dimension ''n''. The domain of M<sub>''f''</sub> consists of vector-valued functions ''ψ'' on '''R''' such that
: <math>\int_\mathbf{R} |\lambda|^2\ \|\psi(\lambda)\|^2 \, d\mu(\lambda) < \infty.</math>

Non-negative countably additive measures ''μ'', ''ν'' are '''mutually singular''' if and only if they are supported on disjoint Borel sets.

{{math theorem|math_statement=Let ''A'' be a self-adjoint operator on a ''separable'' Hilbert space ''H''.  Then there is an ''ω'' sequence of countably additive finite measures on '''R''' (some of which may be identically 0)
<math display="block">\left\{\mu_\ell\right\}_{1 \leq \ell \leq \omega}</math>
such that the measures are pairwise singular and ''A'' is unitarily equivalent to the operator of multiplication by the function ''f''(''λ'') = ''λ'' on
<math display="block">\bigoplus_{1 \leq \ell \leq \omega} L^2_{\mu_\ell} \left(\mathbf{R}, \mathbf{H}_\ell \right).</math>}}

This representation is unique in the following sense: For any two such representations of the same ''A'', the corresponding measures are equivalent in the sense that they have the same sets of measure 0.

=== 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>

=== Example: structure of the Laplacian ===
The Laplacian on '''R'''<sup>''n''</sup> is the operator
: <math>\Delta = \sum_{i=1}^n \partial_{x_i}^2.</math>

As remarked above, the Laplacian is diagonalized by the Fourier transform. Actually it is more natural to consider the ''negative'' of the Laplacian −Δ since as an operator it is non-negative; (see [[elliptic operator]]).

{{math theorem|math_statement=If ''n'' = 1, then −Δ has uniform multiplicity <math>\text{mult} = 2</math>, otherwise −Δ has uniform multiplicity <math>\text{mult} = \omega</math>.  Moreover, the measure ''μ''<sub>'''mult'''</sub> may be taken to be Lebesgue measure on [0, ∞).}}