# spectral measure

## 1 Definition

A spectral measure in $X$ is a function $P:\mathcal{B}\longrightarrow B(H)$ such that

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

• b) $P(\emptyset)=0$.

• c) $P(X)=I$, where $I$ denotes the identity operator  in $B(H)$.

• d) If $E_{1}$ and $E_{2}$ are disjoint subsets of $\mathcal{B}$, then $P(E_{1})$ and $P(E_{2})$ are orthogonal.

• e) $\displaystyle P(\bigcup_{n=1}^{\infty}E_{n})=\sum_{n=1}^{\infty}P(E_{n})\;\;$ for every sequence $E_{1},E_{2},\ldots$ of disjoint sets in $\mathcal{B}$.

The in the last condition is interpreted as the pointwise  limit of the partial sums. Since from condition (d) the projections $P(E_{1}),P(E_{2}),\ldots$ 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  $T\in B(H)$.

• $E_{1}\subseteq E_{2}\;\Longrightarrow\;\mathrm{Ran}(P(E_{1}))\subseteq\mathrm{% Ran}(P(E_{2}))$.

• $P(E_{1}\cap E_{2})=P(E_{1})P(E_{2})\;\;$ 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.

## 3 Equivalent Definition

Theorem 1 - A function $P:\mathcal{B}\longrightarrow B(H)$ whose values are projections is a spectral measure in $X$ if and only if $P(X)=I$ and for every $\xi,\eta\in H$ the function $\mu_{\xi,\eta}:X\longrightarrow\mathbb{C}$ given by

 $\displaystyle\mu_{\xi,\eta}(E):=\langle P(E)\xi,\eta\rangle$

is a complex measure in $X$.

## 4 Integration against spectral measures

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

 $\displaystyle\sum_{i}\lambda_{i}\,P(E_{i})\,,\qquad\qquad\lambda_{i}\in\mathbb% {C}$

Theorem 2 - There exists a unique operator $S\in B(H)$ with the following property: for any given $\epsilon>0$ and for every measurable partition $\{E_{1},\cdots,E_{n}\}$ of $X$ that satisfies $|f(x)-f(x^{\prime})|<\epsilon$ for all $x,x^{\prime}\in E_{i}$, we have

 $\displaystyle\|S-\sum_{i=1}^{n}f(x_{i})P(E_{i})\|<\epsilon$

for any choice of points $x_{i}\in E_{i}$.

$\,$

We can then define $\displaystyle\int_{X}fdP$ 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

 $\mu_{\xi,\eta}(E)=\langle P(E)\,\xi,\eta\rangle,$

whose total variation   is estimated by $\|\mu_{\xi,\eta}\|\leq\|\xi\|\|\eta\|$.

Then we notice that the function $[\cdot,\cdot]:H\times H\longrightarrow\mathbb{C}$ defined by

 $[\xi,\eta]:=\int_{X}f\;d\mu_{\xi,\eta}$

Then, by the Riesz representation theorem (http://planetmath.org/RieszRepresentationTheoremOfBoundedSesquilinearForms), there exists a unique operator $S\in B(H)$ such that

 $\displaystyle\langle S\xi,\eta\rangle=\int_{X}f\;d\mu_{\xi,\eta},\quad\quad\xi% ,\eta\in H$ (1)

We can then define $\displaystyle\int_{X}f\;dP$ 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:

• $\displaystyle\big{\langle}\int_{X}f\;dP\,\xi,\eta\big{\rangle}=\int_{X}f\;d\mu% _{\xi,\eta}$

• $\displaystyle\int_{X}f\;dP$ can be arbitrarilly approximated in norm by operators of the form $\displaystyle\sum_{i=1}^{n}f(x_{i})P(E_{i})$.

## 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 $\mathbb{C}$ supported in a compact set arise from a normal operator. Thus, to any such spectral projection we can associate a normal operator and vice-versa. This interplay between spectral projections and normal operators is deeply explored in some versions of the spectral theorem  .

## References

 Title spectral measure Canonical name SpectralMeasure Date of creation 2013-03-22 17:32:06 Last modified on 2013-03-22 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