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 $(X,ℬ)$ a measurable space.

Definition - A spectral measure in $X$ is a function $P\mathrm{:}\mathrm{B}\mathrm{⟶}B\mathit{}\mathrm{(}H\mathrm{)}$ 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 $T\in B(H)$.

• $E_1\subseteq E_2⟹\mathrm{Ran}(P(E_1))\subseteq \mathrm{Ran}(P(E_2))$.

• $P(E_1\cap E_2)=P(E_1)P(E_2)$ for every $E_1,E_2\in ℬ$.

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

• Let $(X,ℬ,\mu )$ 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

  $P:X$ $⟶B(L^2(X,\mu ))$

  $P(E)$ $:={\chi }_E$

• 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:

  $P:X$ $⟶B(H)$

  $P(E)$ $:={\chi }_E(T)$

3 Equivalent Definition

The following result provides a very useful equivalent definition of a spectral measure.

Theorem 1 - A function $P\mathrm{:}\mathrm{B}\mathrm{⟶}B\mathit{}\mathrm{(}H\mathrm{)}$ 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 ${\mu }_{\xi \mathrm{,}\eta }\mathrm{:}X\mathrm{⟶}\mathrm{C}$ given by

${\mu }_{\xi ,\eta }(E):=⟨P(E)\xi ,\eta ⟩$

is a complex measure in $X$.

4 Integration against spectral measures

Let $f:X⟶ℂ$ 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

$${\int }_{X}f𝑑P$$

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

${\displaystyle \sum _i}{\lambda }_{i}P(E_i),{\lambda }_i\in ℂ$

Theorem 2 - There exists a unique operator $S\mathrm{\in }B\mathit{}\mathrm{(}H\mathrm{)}$ with the following property: for any given $ϵ\mathrm{>}\mathrm{0}$ 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 ${\displaystyle {\int }_X}f𝑑P$ 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

$${\mu }_{\xi ,\eta }(E)=⟨P(E)\xi ,\eta ⟩,$$

whose total variation is estimated by $\parallel {\mu }_{\xi ,\eta }\parallel \le \parallel \xi \parallel \parallel \eta \parallel$.

Then we notice that the function $[\cdot ,\cdot ]:H\times H⟶ℂ$ defined by

$$[\xi ,\eta ]:={\int }_{X}f𝑑{\mu }_{\xi ,\eta }$$

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

$⟨S\xi ,\eta ⟩={\displaystyle {\int }_X}f𝑑{\mu }_{\xi ,\eta },\xi ,\eta \in H$ (1)

We can then define ${\displaystyle {\int }_X}f𝑑P$ 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:

• $⟨{\displaystyle {\int }_X}f𝑑P\xi ,\eta ⟩={\displaystyle {\int }_X}f𝑑{\mu }_{\xi ,\eta }$

• ${\displaystyle {\int }_X}f𝑑P$ 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 $ℂ$ 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.

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