Editing Self-adjoint operator (section)

=== Multiplication operator form of the spectral theorem ===
Firstly, let <math>(X, \Sigma, \mu)</math> be a [[Measure space#Important classes of measure spaces|σ-finite measure space]] and <math>h : X \to \mathbb{R}</math> a [[measurable function]] on <math>X</math>. Then the operator <math>T_h : \operatorname{Dom}T_h \to L^2(X,\mu)</math>, defined by
: <math>T_h \psi(x) = h(x)\psi(x), \quad \forall \psi \in \operatorname{Dom}T_h,</math>
where
: <math>\operatorname{Dom}T_h := \left\{\psi\in L^2(X,\mu) \;|\; h\psi \in L^2(X,\mu)\right\},</math>
is called a '''[[multiplication operator]]'''.{{sfn|Hall|2013|p=207|ps=}} Any multiplication operator is a self-adjoint operator.{{sfn | Akhiezer | 1981 | p=152|ps=}}

Secondly, two operators <math>A</math> and <math>B</math> with dense domains <math>\operatorname{Dom}A \subseteq H_1</math> and <math>\operatorname{Dom}B \subseteq H_2</math> in Hilbert spaces <math>H_1</math> and <math>H_2</math>, respectively, are '''unitarily equivalent''' if and only if there is a [[unitary transformation]] <math>U: H_1 \to H_2</math> such that:{{sfn | Akhiezer | 1981 | pp=115-116|ps=}}
* <math>U\operatorname{Dom}A = \operatorname{Dom}B,</math>
* <math> U A U^{-1} \xi = B \xi, \quad \forall \xi \in \operatorname{Dom}B. </math>
If unitarily equivalent <math>A</math> and <math>B</math> are bounded, then <math>\|A\|_{H_1}=\|B\|_{H_2}</math>; if <math>A</math> is self-adjoint, then so is <math>B</math>. 

{{math theorem|Any self-adjoint operator <math>A</math> on a [[separable space|separable]] Hilbert space is unitarily equivalent to a multiplication operator, i.e.,{{sfn|Hall|2013|pp=127,207|ps=}}
: <math>UAU^{-1}\psi(x) = h(x)\psi(x), \quad \forall \psi \in U\operatorname{Dom}(A)</math>
}}
The spectral theorem holds for both bounded and unbounded self-adjoint operators. Proof of the latter follows by reduction to the spectral theorem for [[unitary operator]]s.<ref>{{harvnb|Hall|2013}} Section 10.4</ref> We might note that if <math>T</math> is multiplication by <math>h</math>, then the spectrum of <math>T</math> is just the [[essential range]] of <math>h</math>.

More complete versions of the spectral theorem exist as well that involve direct integrals and carry with it the notion of "generalized eigenvectors".{{sfn|Hall|2013|pp=144-147,206-207|ps=}}