Editing Self-adjoint operator (section)

=== 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.