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 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 satisfying the following conditions:
(monotony): For with we have .
(continuity from above): for all we have .
(boundary condition): and .
If there is a finite such that for all and for all , than the resolution is called bounded, otherwise unbounded.
spectral measure and spectral integral
Let E be a spectral resolution and be a bounded interval in . The spectral measure of with respect to is given by
This allows us to define the integral of a step function 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 need not be a bounded operator of course, the domain of 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 be a locally compact, abelian topological group, the character group of , a Hilbert space and an unitary representation of 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 on such that:
The equality holds in the weak sense, i.e. the integral converges in the weak operator topology. The spectrum of , denoted by , is defined to be the support of this spectral measure .
The Case of the Translation Group
The groups of translations on is both isomorph to and to it’s own character group, every character is of the form for a fixed . So in this case theorem 1 becomes:
This allows us to talk about the support of the spectral measure, i.e. the spectrum of , as a subset of .
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