Contents

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 $ℬ(\mathscr{H})$ be the algebra of bounded linear operators on $ℍ$ and $𝒫(\mathscr{H})$ the orthogonal projections.

real spectral measure

The following paragraphs will explain the concept of a spectral measure in the real case, this is sufficient if one is interested in spectral theorems of selfadjoint operators only, and can also serve as a simple introduction to the general ideas.

resolution of identity

Do not confuse this concept with the “resolution of identity” in (real) differential geometry.

definition: A resolution of the identity is a map $E:ℝ\to 𝒫(\mathscr{H})$ satisfying the following conditions:

1. (monotony): For ${\lambda }_1,{\lambda }_2\in ℝ$ with ${\lambda }_1\le {\lambda }_2$ we have $E({\lambda }_1)\le E({\lambda }_2)$.

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

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

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 Intervall in $ℝ$. We define the spectral measure of $I$ with respect to $E$ as

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

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

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

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

The equality holds in the weak sense, i.e. the integral converges in the weak operator topology. The spectrum of $𝒰(𝒢)$, denoted by $\mathrm{spec}𝒰(𝒢)$, 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 $a\mapsto \exp (i⟨a,k⟩)$ for a fixed $k\in R^n$. So in this case theorem 1 becomes:

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

References

The theorem 1 is theorem 4.44 in the following classic book:

• Folland, Gerald B.: A course in abstract harmonic analysis. CRC Press 1995 (ZMATH entry).