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 $\mathcal{B}(\mathscr{H})$ be the algebra of bounded linear operators on $\mathbb{H}$ and $\mathcal{P}(\mathscr{H})$ the orthogonal projections.
The following paragraphs explain the concept of a spectral measure in the real case, sufficient for spectral theorems of selfadjoint operators.
Do not confuse this concept with the partition of unity in differential geometry.
definition: A resolution of the identity operator is a map $E:\mathbb{R}\to \mathcal{P}(\mathscr{H})$ satisfying the following conditions:
(monotony): For ${\lambda}_{1},{\lambda}_{2}\in \mathbb{R}$ with ${\lambda}_{1}\le {\lambda}_{2}$ we have $E({\lambda}_{1})\le E({\lambda}_{2})$.
(continuity from above): for all $\lambda \in \mathbb{R}$ we have $s-{\mathrm{lim}}_{\u03f5\to 0,\u03f5>0}E(\lambda +\u03f5)=E(\lambda )$.
(boundary condition): $s-{\mathrm{lim}}_{\u03f5\to -\mathrm{\infty}}E(\lambda )=0$ and $s-{\mathrm{lim}}_{\u03f5\to \mathrm{\infty}}E(\lambda )=\U0001d7d9$.
If there is a finite $\mu \in \mathbb{R}$ such that ${E}_{\lambda}=0$ for all $\lambda \le \mu $ and ${E}_{\lambda}=\U0001d7d9$ for all $\lambda \ge \mu $, than the resolution is called bounded, otherwise unbounded.
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
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):
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 $\mathcal{G}$, $\mathscr{H}$ a Hilbert space and $\mathcal{U}$ an unitary representation of $\mathcal{G}$ 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 $\mathcal{U}(\mathcal{G})$, denoted by $\mathrm{spec}\mathcal{U}(\mathcal{G})$, is defined to be the support of this spectral measure $\mathcal{P}$.
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 $a\mapsto \mathrm{exp}(i\u27e8a,k\u27e9)$ 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 $\mathcal{U}(\mathcal{T})$, as a subset of ${R}^{n}$.
See also projection measure. The theorem 1 is theorem 4.44 in the following classic book: