Editing Projection-valued measure (section)

== Extensions of projection-valued measures ==

If {{pi}} is a projection-valued measure on a measurable space (''X'', ''M''), then the map

: <math>
 \chi_E \mapsto \pi(E)
</math>

extends to a linear map on the vector space of [[step function]]s on ''X''.  In fact, it is easy to check that this map is a [[ring homomorphism]].  This map extends in a canonical way to all bounded complex-valued [[measurable function]]s on ''X'', and we have the following.

{{math theorem|Theorem|For any bounded Borel function <math>f</math> on <math>X</math>, there exists a unique [[bounded operator]] <math> T : H \to H </math> such that
<ref>{{Citation |last=Kowalski|first=Emmanuel| year=2009|title=Spectral theory in Hilbert spaces| series = ETH Zürich lecture notes | url=https://people.math.ethz.ch/~kowalski/spectral-theory.pdf|page = 50}}</ref>{{sfn | Reed | Simon | 1980 | p=227,235}}
:<math>\langle T \xi \mid  \xi \rangle = \int_X f(\lambda) \,d\mu_{\xi}(\lambda), \quad \forall \xi \in H.</math>
where <math>\mu_{\xi}</math> is a finite [[Borel measure]] given by
:<math>\mu_{\xi}(E) := \langle \pi(E)\xi \mid \xi \rangle, \quad \forall E \in M.</math>
Hence, <math>(X,M,\mu)</math> is a [[finite measure space]].}}

The theorem is also correct for unbounded measurable functions <math>f</math> but then <math>T</math> will be an unbounded linear operator on the Hilbert space <math>H</math>.

This allows to define the [[Borel functional calculus]] for such operators and then pass to measurable functions via the [[Riesz–Markov–Kakutani representation theorem]]. That is, if <math>g:\mathbb{R}\to\mathbb{C}</math> is a measurable function, then a unique measure exists such that
:<math>g(T) :=\int_\mathbb{R} g(x) \, d\pi(x).</math>

=== Spectral theorem ===
{{see also|Self-adjoint operator#Spectral theorem}}
Let <math>H</math> be a [[separable space|separable]] [[complex number|complex]] [[Hilbert space]], <math>A:H\to H</math> be a bounded [[self-adjoint operator]] and <math>\sigma(A)</math> the [[Spectrum_(functional_analysis)|spectrum]] of <math>A</math>. Then the [[spectral theorem]] says that there exists a unique projection-valued measure <math>\pi^A</math>, defined on a [[Borel_set|Borel subset]] <math> E \subset \sigma(A)</math>, such that{{sfn | Reed | Simon | 1980 | p=235}}
:<math>A =\int_{\sigma(A)} \lambda \, d\pi^A(\lambda),</math>
where the integral extends to an unbounded function <math>\lambda</math> when the spectrum of <math>A</math> is unbounded.{{sfn | Hall | 2013 | p=205}}

=== Direct integrals===

First we provide a general example of projection-valued measure based on [[direct integral]]s. Suppose (''X'', ''M'', μ) is a measure space and let {''H''<sub>''x''</sub>}<sub>''x'' ∈ ''X'' </sub> be a μ-measurable family of separable Hilbert spaces. For every  ''E'' ∈ ''M'', let {{pi}}(''E'') be the operator of multiplication by 1<sub>''E''</sub> on the Hilbert space

:<math> \int_X^\oplus H_x \ d \mu(x). </math>

Then {{pi}} is a projection-valued measure on (''X'', ''M'').

Suppose {{pi}}, ρ  are projection-valued measures on  (''X'', ''M'') with values in the projections of ''H'', ''K''.   {{pi}},  ρ  are '''unitarily equivalent''' [[if and only if]] there is a unitary operator ''U'':''H'' → ''K'' such that

:<math> \pi(E) = U^* \rho(E) U \quad </math>

for every ''E'' ∈ ''M''.

'''Theorem'''. If (''X'', ''M'') is a [[Borel algebra#Standard Borel spaces and Kuratowski theorems|standard Borel space]], then for every projection-valued measure {{pi}} on (''X'', ''M'') taking values in the projections of a ''separable'' Hilbert space, there is a Borel measure μ and a μ-measurable family of Hilbert spaces {''H''<sub>''x''</sub>}<sub>''x'' ∈ ''X'' </sub>, such that {{pi}} is unitarily equivalent to multiplication by 1<sub>''E''</sub> on the Hilbert space

:<math> \int_X^\oplus H_x \ d \mu(x). </math>

The measure class{{clarify|reason=What is a measure class? A measure up to measure-preserving equivalence? Should the measure be completed?|date=May 2015}} of μ and the measure equivalence class of the multiplicity function ''x'' → dim ''H''<sub>''x''</sub> completely characterize the projection-valued measure up to unitary equivalence.

A projection-valued measure {{pi}} is ''homogeneous of multiplicity'' ''n'' if and only if the multiplicity function has constant value ''n''.  Clearly,

'''Theorem'''.  Any projection-valued measure {{pi}} taking values in the projections of a separable Hilbert space is an orthogonal direct sum of homogeneous projection-valued measures:

:<math> \pi = \bigoplus_{1 \leq n \leq \omega} (\pi \mid H_n) </math>

where

:<math> H_n = \int_{X_n}^\oplus H_x \ d (\mu \mid X_n) (x) </math>

and

:<math> X_n = \{x \in X: \dim H_x = n\}. </math>