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\begin{document}

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\section*{spectral measure}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{contents}{Contents}\dotfill \pageref*{contents} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{real_spectral_measure}{real spectral measure}\dotfill \pageref*{real_spectral_measure} \linebreak
\noindent\hyperlink{resolution_of_identity}{resolution of identity}\dotfill \pageref*{resolution_of_identity} \linebreak
\noindent\hyperlink{spectral_measure_and_spectral_integral}{spectral measure and spectral integral}\dotfill \pageref*{spectral_measure_and_spectral_integral} \linebreak
\noindent\hyperlink{spectrum_of_representations_of_groups_the_snag_theorem}{Spectrum of Representations of Groups, the SNAG Theorem}\dotfill \pageref*{spectrum_of_representations_of_groups_the_snag_theorem} \linebreak
\noindent\hyperlink{the_case_of_the_translation_group}{The Case of the Translation Group}\dotfill \pageref*{the_case_of_the_translation_group} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\section*{{Idea}}\label{idea}

Spectral measures are an essential tool of [[functional analysis]] on [[Hilbert space]]s. Spectral measures are projection-valued measures and are used to state various forms of [[spectral theorem]]s.

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.

\hypertarget{real_spectral_measure}{}\subsection*{{real spectral measure}}\label{real_spectral_measure}

The following paragraphs explain the concept of a spectral measure in the real case, sufficient for [[spectral theorem]]s of [[selfadjoint operator]]s.

\hypertarget{resolution_of_identity}{}\subsubsection*{{resolution of identity}}\label{resolution_of_identity}

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

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

\begin{enumerate}%
\item (\textbf{monotony}): For $\lambda_1, \lambda_2 \in \mathbb{R}$ with $\lambda_1 \leq \lambda_2$ we have $E(\lambda_1) \leq E(\lambda_2)$.


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


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



\end{enumerate}
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 \textbf{bounded}, otherwise \textbf{unbounded}.

\hypertarget{spectral_measure_and_spectral_integral}{}\subsubsection*{{spectral measure and spectral integral}}\label{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

\begin{displaymath}
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}
\end{displaymath}
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

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

\begin{displaymath}
E(u) = \integral u(\lambda) dE(\lambda)
\end{displaymath}
For general function $u, E(u)$ need not be a bounded operator of course, the domain of $E(u)$ is (theorem):

\begin{displaymath}
D(E(u)) = \{ f \in \mathcal{H} : \int |u(\lambda)|^2 d\langle E(\lambda)f, f\rangle \lt \infty \}
\end{displaymath}
\hypertarget{spectrum_of_representations_of_groups_the_snag_theorem}{}\section*{{Spectrum of Representations of Groups, the SNAG Theorem}}\label{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) \textbf{SNAG} theorem (SNAG = Stone-Naimark-Ambrose-Godement):

\begin{theorem}
\label{specmeasure}\hypertarget{specmeasure}{}
\begin{displaymath}
\mathcal{U}(g) = \int_{\chi\in\hat \mathcal{G}} \langle g, \chi\rangle \mathcal{P}(d\chi) \qquad \forall g \in \mathcal{G}
\end{displaymath}
\end{theorem}
The equality holds in the weak sense, i.e. the integral converges in the weak operator topology. The \textbf{spectrum} of $\mathcal{U}(\mathcal{G})$, denoted by $spec\mathcal{U}(\mathcal{G})$, is defined to be the support of this spectral measure $\mathcal{P}$.

\hypertarget{the_case_of_the_translation_group}{}\subsection*{{The Case of the Translation Group}}\label{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 \ref{specmeasure} becomes:

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

\hypertarget{references}{}\subsection*{{References}}\label{references}

See also [[projection measure]]. The theorem \ref{specmeasure} is theorem 4.44 in the following classic book:

\begin{itemize}%
\item Gerald B. Folland, \emph{A course in abstract harmonic analysis}, Studies in Adv. Math. CRC Press 1995, \href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0857.43001&format=complete}{Zbl}
\item A. A. Kirillov, A. D. Gvi\v{s}iani,      (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988

\end{itemize}
[[!redirects spectral measures]]



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