spectral measure



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 𝒫()\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:𝒫()E: \mathbb{R} \to \mathcal{P}(\mathcal{H}) satisfying the following conditions:

  1. (monotony): For λ 1,λ 2\lambda_1, \lambda_2 \in \mathbb{R} with λ 1λ 2\lambda_1 \leq \lambda_2 we have E(λ 1)E(λ 2)E(\lambda_1) \leq E(\lambda_2).

  2. (continuity from above): for all λ\lambda \in \mathbb{R} we have slim ϵ0,ϵ>0E(λ+ϵ)=E(λ)s-\lim_{\epsilon \to 0, \epsilon \gt 0} E(\lambda + \epsilon) = E(\lambda).

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

If there is a finite μ\mu \in \mathbb{R} such that E λ=0E_{\lambda} = 0 for all λμ\lambda \leq \mu and E λ=𝟙E_{\lambda} = \mathbb{1} for all λμ\lambda \geq \mu, than the resolution is called bounded, otherwise unbounded.

spectral measure and spectral integral

Let E be a spectral resolution and II be a bounded interval in \mathbb{R}. The spectral measure of II with respect to EE is given by

E(J):={E(y)E(x) for I=(x,y) E(y)E(x) for I=[x,y) E(y)E(x) for I=(x,y] E(y)E(x) for I=[x,y] 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= k=1 nα kχ I ku = \sum_{k=1}^{n} \alpha_k \chi_{I_k} with respect to E as

u(λ)dE(λ):= k=1 nα kE(I k) \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)=u(λ)dE(λ) E(u) = \integral u(\lambda) dE(\lambda)

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

D(E(u))={f:|u(λ)| 2dE(λ)f,f<} 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\mathcal{G} be a locally compact, abelian topological group, 𝒢^\hat \mathcal{G} the character group of 𝒢\mathcal{G}, \mathcal{H} a Hilbert space and 𝒰\mathcal{U} an unitary representation of 𝒢\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 𝒫\mathcal{P} on 𝒢^\hat \mathcal{G} such that:
𝒰(g)= χ𝒢^g,χ𝒫(dχ)g𝒢 \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(𝒢)\mathcal{U}(\mathcal{G}), denoted by spec𝒰(𝒢)spec\mathcal{U}(\mathcal{G}), is defined to be the support of this spectral measure 𝒫\mathcal{P}.

The Case of the Translation Group

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

𝒰(t)= kR ne it,k𝒫(k)t𝒯 \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(𝒯)\mathcal{U}(\mathcal{T}), as a subset of R n\R^n.


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