spectral measure

1 Definition

In this entry by a projection we an orthogonal projection over some Hilbert spaceMathworldPlanetmath. Also, we say that two projections are orthogonalMathworldPlanetmathPlanetmath if their images are orthogonal subspaces.

Let H be an Hilbert space, B(H) the algebra of bounded operatorsMathworldPlanetmathPlanetmath in H and (X,) a measurable spaceMathworldPlanetmathPlanetmath.

Definition - A spectral measure in X is a function P:BB(H) such that

  • a) P(E) is a projection in B(H) for every EB.

  • b) P()=0.

  • c) P(X)=I, where I denotes the identity operatorMathworldPlanetmath in B(H).

  • d) If E1 and E2 are disjoint subsets of B, then P(E1) and P(E2) are orthogonal.

  • e) P(n=1En)=n=1P(En) for every sequence E1,E2, of disjoint sets in B.

The in the last condition is interpreted as the pointwisePlanetmathPlanetmath limit of the partial sums. Since from condition (d) the projections P(E1),P(E2), are orthogonal, we know that the pointwise limit exists and is a projection (see this entry (http://planetmath.org/LatticeOfProjections), TheoremMathworldPlanetmath 5).

: In the following Ran(T) denotes the range (http://planetmath.org/Function) of an operatorMathworldPlanetmath TB(H).

  • E1E2Ran(P(E1))Ran(P(E2)).

  • P(E1E2)=P(E1)P(E2) for every E1,E2.

Thus, a spectral measure is a countably additive vector measure whose values are projections. For that, spectral measures are also called projection valued measures.

2 Examples

  • Let (X,,μ) be a measure spaceMathworldPlanetmath. Consider the Hilbert space L2(X,μ) (http://planetmath.org/L2SpacesAreHilbertSpaces). We regard a function f in L(X,μ) (http://planetmath.org/LpSpace) as the multiplication operator MfB(L2(X,μ)) given by


    In this setting, the characteristic functionsMathworldPlanetmathPlanetmathPlanetmathPlanetmath are projections in B(L2(X,μ)) and we have a spectral measure given by

    P:X B(L2(X,μ))
    P(E) :=χE
  • Let H be a Hilbert space, TB(H) a normal operator and σ(T) the spectrum of T. For any measurable subset Eσ(T) the operators χE(T), given by the Borel functional calculus, are projections in B(H). Moreover, we have a spectral measure given by:

    P:X B(H)
    P(E) :=χE(T)

3 Equivalent Definition

The following result provides a very useful equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath definition of a spectral measure.

Theorem 1 - A function P:BB(H) whose values are projections is a spectral measure in X if and only if P(X)=I and for every ξ,ηH the function μξ,η:XC given by


is a complex measure in X.

4 Integration against spectral measures

Let f:X be a boundedPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Bounded) measurable functionMathworldPlanetmath and P a spectral measure in X. We are interested to give meaning to the integral


Since we are dealing with “measures” whose values are linear operatorsMathworldPlanetmath it is reasonable to expect that this integral is itself a linear operator.

There are two natural ways to define it that turn out to be equivalent. The first approach is a construction that resembles the approximation of f by simple functionsMathworldPlanetmath in Lebesgue integral theory. Here the role of simple functions will be played by the operators of the form


Theorem 2 - There exists a unique operator SB(H) with the following property: for any given ϵ>0 and for every measurable partition {E1,,En} of X that satisfies |f(x)-f(x)|<ϵ for all x,xEi, we have


for any choice of points xiEi.

We can then define Xf𝑑P as the unique operator S described by Theorem 2.

The other approach to define this integral is by specifying an appropriate bounded sesquilinear form. Recall that from Riesz representation theoremMathworldPlanetmath (http://planetmath.org/RieszRepresentationTheoremOfBoundedSesquilinearForms), to every bounded sesquilinear form corresponds a unique bounded operator. The construction is as follows:

First we notice that, from the alternative defintion of spectral measure (Theorem 1), for every vectors ξ,ηH we can define a complex measure μξ,η by


whose total variationMathworldPlanetmathPlanetmath is estimated by μξ,ηξη.

Then we notice that the function [,]:H×H defined by


is a sesquilinear formPlanetmathPlanetmath.

Then, by the Riesz representation theorem (http://planetmath.org/RieszRepresentationTheoremOfBoundedSesquilinearForms), there exists a unique operator SB(H) such that

Sξ,η=Xf𝑑μξ,η,ξ,ηH (1)

We can then define Xf𝑑P as this operator S. Of course, the two definitions are equivalent. We summarize this in the following result

Theorem 3 - Given a spectral measure P and a bounded Borel function f, an operator S that satisfies condition (1) also satisfies the conditions of Theorem 2. Therefore, both definitions of the integral of f with respect to P coincide and we have that:

  • Xf𝑑Pξ,η=Xf𝑑μξ,η

  • Xf𝑑P can be arbitrarilly approximated in norm by operators of the form i=1nf(xi)P(Ei).

5 Remarks

The second example we gave above, of a spectral measure associated with a normal operator, is in some sense the general case: all spectral projections in supported in a compact set arise from a normal operator. Thus, to any such spectral projection we can associate a normal operator and vice-versa. This interplay between spectral projections and normal operators is deeply explored in some versions of the spectral theoremMathworldPlanetmath.


  • 1 W. Arveson, A Short Course on Spectral Theory, Graduate Texts in Mathematics, 209, Springer, New York, 2002
  • 2 J. B. Conway, A Course in Functional AnalysisMathworldPlanetmath, 2nd ed., Graduate Texts in Mathematics, 96, Springer-Verlag, New York, Berlin, 1990.
Title spectral measure
Canonical name SpectralMeasure
Date of creation 2013-03-22 17:32:06
Last modified on 2013-03-22 17:32:06
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 11
Author asteroid (17536)
Entry type Definition
Classification msc 47A56
Classification msc 46G12
Classification msc 46G10
Classification msc 28C20
Classification msc 28B05
Synonym projection valued measure
Defines integration against spectral measures