nLab
spectral measure

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 be a Hilbert space and () be the algebra of bounded linear operators on and 𝒫() 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() satisfying the following conditions:

  1. (monotony): For λ 1,λ 2 with λ 1λ 2 we have E(λ 1)E(λ 2).

  2. (continuity from above): for all λ we have slim ϵ0,ϵ>0E(λ+ϵ)=E(λ).

  3. (boundary condition): slim ϵE(λ)=0 and slim ϵE(λ)=𝟙.

If there is a finite μ such that E λ=0 for all λμ and E λ=𝟙 for all λμ, 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 . The spectral measure of I with respect to E 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 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) need not be a bounded operator of course, the domain of 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 be a locally compact, abelian topological group, 𝒢^ the character group of 𝒢, a Hilbert space and 𝒰 an unitary representation of 𝒢 in the algebra of bounded operators of . The following theorem is sometimes called (classical) SNAG theorem (SNAG = Stone-Naimark-Ambrose-Godement):

  • Theorem: There is a unique regular spectral measure 𝒫 on 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), denoted by spec𝒰(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 aexp(ia,k) for a fixed kR n. So in this case theorem 1 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), as a subset of R n.

References

See also projection measure. The theorem 1 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

Revised on June 4, 2011 12:29:23 by Zoran Škoda (31.45.147.163)