# Idea

Spectral measures are an essential tool of functional analysis on Hilbert spaces. Spectral measures are projection-valued measures and are used to state various forms of spectral theorems.

In the following, let $ℍ$ be a Hilbert space and $ℬ\left(ℋ\right)$ be the algebra of bounded linear operators on $ℍ$ and $𝒫\left(ℋ\right)$ the orthogonal projections.

## real spectral measure

The following paragraphs explain the concept of a spectral measure in the real case, sufficient for spectral theorems of selfadjoint operators.

### resolution of identity

Do not confuse this concept with the partition of unity in differential geometry.

definition: A resolution of the identity operator is a map $E:ℝ\to 𝒫\left(ℋ\right)$ satisfying the following conditions:

1. (monotony): For ${\lambda }_{1},{\lambda }_{2}\in ℝ$ with ${\lambda }_{1}\le {\lambda }_{2}$ we have $E\left({\lambda }_{1}\right)\le E\left({\lambda }_{2}\right)$.

2. (continuity from above): for all $\lambda \in ℝ$ we have $s-{\mathrm{lim}}_{ϵ\to 0,ϵ>0}E\left(\lambda +ϵ\right)=E\left(\lambda \right)$.

3. (boundary condition): $s-{\mathrm{lim}}_{ϵ\to -\infty }E\left(\lambda \right)=0$ and $s-{\mathrm{lim}}_{ϵ\to \infty }E\left(\lambda \right)=𝟙$.

If there is a finite $\mu \in ℝ$ such that ${E}_{\lambda }=0$ for all $\lambda \le \mu$ and ${E}_{\lambda }=𝟙$ for all $\lambda \ge \mu$, than the resolution is called bounded, otherwise unbounded.

### spectral measure and spectral integral

Let E be a spectral resolution and $I$ be a bounded interval in $ℝ$. The spectral measure of $I$ with respect to $E$ is given by

$E\left(J\right):=\left\{\begin{array}{ll}E\left(y-\right)-E\left(x\right)& \text{for}\phantom{\rule{1em}{0ex}}I=\left(x,y\right)\\ E\left(y-\right)-E\left(x-\right)& \text{for}\phantom{\rule{1em}{0ex}}I=\left[x,y\right)\\ E\left(y\right)-E\left(x\right)& \text{for}\phantom{\rule{1em}{0ex}}I=\left(x,y\right]\\ E\left(y\right)-E\left(x-\right)& \text{for}\phantom{\rule{1em}{0ex}}I=\left[x,y\right]\\ \end{array}$E(J):= \begin{cases} E(y-) - E(x) & \text{for }\quad I=(x,y) \\ E(y-) - E(x-) & \text{for }\quad I=[x,y) \\ E(y) - E(x) & \text{for }\quad I=(x,y] \\ E(y) - E(x-) & \text{for }\quad I=[x,y] \\ \end{cases}

This allows us to define the integral of a step function $u={\sum }_{k=1}^{n}{\alpha }_{k}{\chi }_{{I}_{k}}$ with respect to E as

$\int u\left(\lambda \right)\mathrm{dE}\left(\lambda \right):=\sum _{k=1}^{n}{\alpha }_{k}E\left({I}_{k}\right)$\integral u(\lambda) dE(\lambda) := \sum_{k=1}^{n} \alpha_k E(I_k)

The value of this integral is a bounded operator.

As in conventional measure and integration theory, the integral can be extended from step functions to Borel-measurable functions. In this case one often used notation is

$E\left(u\right)=\int u\left(\lambda \right)\mathrm{dE}\left(\lambda \right)$E(u) = \integral u(\lambda) dE(\lambda)

For general function $u,E\left(u\right)$ need not be a bounded operator of course, the domain of $E\left(u\right)$ is (theorem):

$D\left(E\left(u\right)\right)=\left\{f\in ℋ:\int \mid u\left(\lambda \right){\mid }^{2}d⟨E\left(\lambda \right)f,f⟩<\infty \right\}$D(E(u)) = \{ f \in \mathcal{H} : \int |u(\lambda)|^2 d\langle E(\lambda)f, f\rangle \lt \infty \}

# Spectrum of Representations of Groups, the SNAG Theorem

The SNAG theorem is necessary to explain the spectrum condition of the Haag-Kastler axioms.

Let $𝒢$ be a locally compact, abelian topological group, $\stackrel{^}{𝒢}$ the character group of $𝒢$, $ℋ$ a Hilbert space and $𝒰$ an unitary representation of $𝒢$ in the algebra of bounded operators of $ℋ$. The following theorem is sometimes called (classical) SNAG theorem (SNAG = Stone-Naimark-Ambrose-Godement):

• Theorem: There is a unique regular spectral measure $𝒫$ on $\stackrel{^}{𝒢}$ such that:
$𝒰\left(g\right)={\int }_{\chi \in \stackrel{^}{𝒢}}⟨g,\chi ⟩𝒫\left(d\chi \right)\phantom{\rule{2em}{0ex}}\forall g\in 𝒢$\mathcal{U}(g) = \int_{\chi\in\hat \mathcal{G}} \langle g, \chi\rangle \mathcal{P}(d\chi) \qquad \forall g \in \mathcal{G}

The equality holds in the weak sense, i.e. the integral converges in the weak operator topology. The spectrum of $𝒰\left(𝒢\right)$, denoted by $\mathrm{spec}𝒰\left(𝒢\right)$, is defined to be the support of this spectral measure $𝒫$.

## The Case of the Translation Group

The groups of translations $𝒯$ on ${R}^{n}$ is both isomorph to ${R}^{n}$ and to it’s own character group, every character is of the form $a↦\mathrm{exp}\left(i⟨a,k⟩\right)$ for a fixed $k\in {R}^{n}$. So in this case theorem 1 becomes:

$𝒰\left(t\right)={\int }_{k\in {R}^{n}}{e}^{i⟨t,k⟩}𝒫\left(k\right)\phantom{\rule{2em}{0ex}}\forall t\in 𝒯$\mathcal{U}(t) = \int_{k\in \R^n} e^{i \langle t, k\rangle} \mathcal{P}(k) \qquad \forall t \in \mathcal{T}

This allows us to talk about the support of the spectral measure, i.e. the spectrum of $𝒰\left(𝒯\right)$, as a subset of ${R}^{n}$.

## References

See also projection measure. The theorem 1 is theorem 4.44 in the following classic book:

• Gerald B. Folland, A course in abstract harmonic analysis, Studies in Adv. Math. CRC Press 1995, Zbl
• A. A. Kirillov, A. D. Gvišiani, Теоремы и задачи функционального анализа (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988

Revised on June 4, 2011 12:29:23 by Zoran Škoda (31.45.147.163)