# 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 $\mathbb{H}$ be a Hilbert space and $\mathcal{B}(\mathcal{H})$ be the algebra of bounded linear operators on $\mathbb{H}$ and $\mathcal{P}(\mathcal{H})$ 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 $E: \mathbb{R} \to \mathcal{P}(\mathcal{H})$ satisfying the following conditions:

1. (monotony): For $\lambda_1, \lambda_2 \in \mathbb{R}$ with $\lambda_1 \leq \lambda_2$ we have $E(\lambda_1) \leq E(\lambda_2)$.

2. (continuity from above): for all $\lambda \in \mathbb{R}$ we have $s-\lim_{\epsilon \to 0, \epsilon \gt 0} E(\lambda + \epsilon) = E(\lambda)$.

3. (boundary condition): $s-\lim_{\epsilon \to -\infty} E(\lambda) = 0$ and $s-\lim_{\epsilon \to \infty} E(\lambda) = \mathbb{1}$.

If there is a finite $\mu \in \mathbb{R}$ such that $E_{\lambda} = 0$ for all $\lambda \leq \mu$ and $E_{\lambda} = \mathbb{1}$ for all $\lambda \geq \mu$, 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 $\mathbb{R}$. The spectral measure of $I$ with respect to $E$ is given by

$E(J):= \begin{cases} E(y-) - E(x) & \text{for }\quad I=(x,y) \\ E(y-) - E(x-) & \text{for }\quad I=[x,y) \\ E(y) - E(x) & \text{for }\quad I=(x,y] \\ E(y) - E(x-) & \text{for }\quad I=[x,y] \\ \end{cases}$

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

$\integral u(\lambda) dE(\lambda) := \sum_{k=1}^{n} \alpha_k E(I_k)$

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

$E(u) = \integral u(\lambda) dE(\lambda)$

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

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

# Spectrum of Representations of Groups, the SNAG Theorem

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

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

• Theorem: There is a unique regular spectral measure $\mathcal{P}$ on $\hat \mathcal{G}$ such that:
$\mathcal{U}(g) = \int_{\chi\in\hat \mathcal{G}} \langle g, \chi\rangle \mathcal{P}(d\chi) \qquad \forall g \in \mathcal{G}$

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

## The Case of the Translation Group

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

$\mathcal{U}(t) = \int_{k\in \R^n} e^{i \langle t, k\rangle} \mathcal{P}(k) \qquad \forall t \in \mathcal{T}$

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

See also projection measure. The theorem 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