spectral measure
1 Definition
In this entry by a projection we an orthogonal projection over some Hilbert space^{}. Also, we say that two projections are orthogonal^{} if their images are orthogonal subspaces.
Let $H$ be an Hilbert space, $B(H)$ the algebra of bounded operators^{} in $H$ and $(X,\mathcal{B})$ a measurable space^{}.
Definition  A spectral measure in $X$ is a function $P\mathrm{:}\mathrm{B}\mathrm{\u27f6}B\mathit{}\mathrm{(}H\mathrm{)}$ such that

a) $P\mathit{}\mathrm{(}E\mathrm{)}$ is a projection in $B\mathit{}\mathrm{(}H\mathrm{)}$ for every $E\mathrm{\in}\mathrm{B}$.

b) $P\mathit{}\mathrm{(}\mathrm{\varnothing}\mathrm{)}\mathrm{=}\mathrm{0}$.

c) $P\mathit{}\mathrm{(}X\mathrm{)}\mathrm{=}I$, where $I$ denotes the identity operator^{} in $B\mathit{}\mathrm{(}H\mathrm{)}$.

d) If ${E}_{\mathrm{1}}$ and ${E}_{\mathrm{2}}$ are disjoint subsets of $\mathrm{B}$, then $P\mathit{}\mathrm{(}{E}_{\mathrm{1}}\mathrm{)}$ and $P\mathit{}\mathrm{(}{E}_{\mathrm{2}}\mathrm{)}$ are orthogonal.

e) $P\mathit{}\mathrm{(}{\displaystyle \mathrm{\bigcup}_{n\mathrm{=}\mathrm{1}}^{\mathrm{\infty}}}{E}_{n}\mathrm{)}\mathrm{=}{\displaystyle \mathrm{\sum}_{n\mathrm{=}\mathrm{1}}^{\mathrm{\infty}}}P\mathit{}\mathrm{(}{E}_{n}\mathrm{)}$ for every sequence ${E}_{\mathrm{1}}\mathrm{,}{E}_{\mathrm{2}}\mathrm{,}\mathrm{\dots}$ of disjoint sets in $\mathrm{B}$.
The in the last condition is interpreted as the pointwise^{} limit of the partial sums. Since from condition (d) the projections $P({E}_{1}),P({E}_{2}),\mathrm{\dots}$ are orthogonal, we know that the pointwise limit exists and is a projection (see this entry (http://planetmath.org/LatticeOfProjections), Theorem^{} 5).
$$
: In the following $\mathrm{Ran}(T)$ denotes the range (http://planetmath.org/Function) of an operator^{} $T\in B(H)$.

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${E}_{1}\subseteq {E}_{2}\u27f9\mathrm{Ran}(P({E}_{1}))\subseteq \mathrm{Ran}(P({E}_{2}))$.

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$P({E}_{1}\cap {E}_{2})=P({E}_{1})P({E}_{2})$ for every ${E}_{1},{E}_{2}\in \mathcal{B}$.
Thus, a spectral measure is a countably additive vector measure whose values are projections. For that, spectral measures are also called projection valued measures.
2 Examples

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Let $(X,\mathcal{B},\mu )$ be a measure space^{}. Consider the Hilbert space ${L}^{2}(X,\mu )$ (http://planetmath.org/L2SpacesAreHilbertSpaces). We regard a function $f$ in ${L}^{\mathrm{\infty}}(X,\mu )$ (http://planetmath.org/LpSpace) as the multiplication operator ${M}_{f}\in B({L}^{2}(X,\mu ))$ given by
${M}_{f}(\xi )=f\xi ,\xi \in {L}^{2}(X,\mu )$ In this setting, the characteristic functions^{} are projections in $B({L}^{2}(X,\mu ))$ and we have a spectral measure given by
$P:X$ $\u27f6B({L}^{2}(X,\mu ))$ $P(E)$ $:={\chi}_{E}$ 
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Let $H$ be a Hilbert space, $T\in B(H)$ a normal operator and $\sigma (T)$ the spectrum of $T$. For any measurable subset $E\subseteq \sigma (T)$ the operators ${\chi}_{E}(T)$, given by the Borel functional calculus, are projections in $B(H)$. Moreover, we have a spectral measure given by:
$P:X$ $\u27f6B(H)$ $P(E)$ $:={\chi}_{E}(T)$
3 Equivalent Definition
The following result provides a very useful equivalent^{} definition of a spectral measure.
Theorem 1  A function $P\mathrm{:}\mathrm{B}\mathrm{\u27f6}B\mathit{}\mathrm{(}H\mathrm{)}$ whose values are projections is a spectral measure in $X$ if and only if $P\mathit{}\mathrm{(}X\mathrm{)}\mathrm{=}I$ and for every $\xi \mathrm{,}\eta \mathrm{\in}H$ the function ${\mu}_{\xi \mathrm{,}\eta}\mathrm{:}X\mathrm{\u27f6}\mathrm{C}$ given by
${\mu}_{\xi ,\eta}(E):=\u27e8P(E)\xi ,\eta \u27e9$ 
is a complex measure in $X$.
4 Integration against spectral measures
Let $f:X\u27f6\u2102$ be a bounded^{} (http://planetmath.org/Bounded) measurable function^{} and $P$ a spectral measure in $X$. We are interested to give meaning to the integral
$${\int}_{X}f\mathit{d}P$$ 
Since we are dealing with “measures” whose values are linear operators^{} it is reasonable to expect that this integral is itself a linear operator.
There are two natural ways to define it that turn out to be equivalent. The first approach is a construction that resembles the approximation of $f$ by simple functions^{} in Lebesgue integral theory. Here the role of simple functions will be played by the operators of the form
$\sum _{i}}{\lambda}_{i}P({E}_{i}),{\lambda}_{i}\in \u2102$ 
Theorem 2  There exists a unique operator $S\mathrm{\in}B\mathit{}\mathrm{(}H\mathrm{)}$ with the following property: for any given $\u03f5\mathrm{>}\mathrm{0}$ and for every measurable partition $\mathrm{\{}{E}_{\mathrm{1}}\mathrm{,}\mathrm{\cdots}\mathrm{,}{E}_{n}\mathrm{\}}$ of $X$ that satisfies $$ for all $x\mathrm{,}{x}^{\mathrm{\prime}}\mathrm{\in}{E}_{i}$, we have
$$ 
for any choice of points ${x}_{i}\mathrm{\in}{E}_{i}$.
$$
We can then define ${\int}_{X}}f\mathit{d}P$ as the unique operator $S$ described by Theorem 2.
The other approach to define this integral is by specifying an appropriate bounded sesquilinear form. Recall that from Riesz representation theorem^{} (http://planetmath.org/RieszRepresentationTheoremOfBoundedSesquilinearForms), to every bounded sesquilinear form corresponds a unique bounded operator. The construction is as follows:
First we notice that, from the alternative defintion of spectral measure (Theorem 1), for every vectors $\xi ,\eta \in H$ we can define a complex measure ${\mu}_{\xi ,\eta}$ by
$${\mu}_{\xi ,\eta}(E)=\u27e8P(E)\xi ,\eta \u27e9,$$ 
whose total variation^{} is estimated by $\parallel {\mu}_{\xi ,\eta}\parallel \le \parallel \xi \parallel \parallel \eta \parallel $.
Then we notice that the function $[\cdot ,\cdot ]:H\times H\u27f6\u2102$ defined by
$$[\xi ,\eta ]:={\int}_{X}f\mathit{d}{\mu}_{\xi ,\eta}$$ 
is a sesquilinear form^{}.
Then, by the Riesz representation theorem (http://planetmath.org/RieszRepresentationTheoremOfBoundedSesquilinearForms), there exists a unique operator $S\in B(H)$ such that
$\u27e8S\xi ,\eta \u27e9={\displaystyle {\int}_{X}}f\mathit{d}{\mu}_{\xi ,\eta},\xi ,\eta \in H$  (1) 
We can then define ${\int}_{X}}f\mathit{d}P$ as this operator $S$. Of course, the two definitions are equivalent. We summarize this in the following result
$$
Theorem 3  Given a spectral measure $P$ and a bounded Borel function $f$, an operator $S$ that satisfies condition (1) also satisfies the conditions of Theorem 2. Therefore, both definitions of the integral of $f$ with respect to $P$ coincide and we have that:

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$\u27e8{\displaystyle {\int}_{X}}f\mathit{d}P\xi ,\eta \u27e9={\displaystyle {\int}_{X}}f\mathit{d}{\mu}_{\xi ,\eta}$

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${\int}_{X}}f\mathit{d}P$ can be arbitrarilly approximated in norm by operators of the form $\sum _{i=1}^{n}}f({x}_{i})P({E}_{i})$.
5 Remarks
The second example we gave above, of a spectral measure associated with a normal operator, is in some sense the general case: all spectral projections in $\u2102$ supported in a compact set arise from a normal operator. Thus, to any such spectral projection we can associate a normal operator and viceversa. This interplay between spectral projections and normal operators is deeply explored in some versions of the spectral theorem^{}.
References
 1 W. Arveson, A Short Course on Spectral Theory, Graduate Texts in Mathematics, 209, Springer, New York, 2002
 2 J. B. Conway, A Course in Functional Analysis^{}, 2nd ed., Graduate Texts in Mathematics, 96, SpringerVerlag, New York, Berlin, 1990.
Title  spectral measure 
Canonical name  SpectralMeasure 
Date of creation  20130322 17:32:06 
Last modified on  20130322 17:32:06 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  11 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 47A56 
Classification  msc 46G12 
Classification  msc 46G10 
Classification  msc 28C20 
Classification  msc 28B05 
Synonym  projection valued measure 
Defines  integration against spectral measures 