spectral measure
1 Definition
In this entry by a projection we an orthogonal projection over some Hilbert space. Also, we say that two projections are orthogonal if their images are orthogonal subspaces.
Let be an Hilbert space, the algebra of bounded operators in and a measurable space.
Definition - A spectral measure in is a function such that
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a) is a projection in for every .
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b) .
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c) , where denotes the identity operator in .
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d) If and are disjoint subsets of , then and are orthogonal.
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e) for every sequence of disjoint sets in .
The in the last condition is interpreted as the pointwise limit of the partial sums. Since from condition (d) the projections are orthogonal, we know that the pointwise limit exists and is a projection (see this entry (http://planetmath.org/LatticeOfProjections), Theorem 5).
: In the following denotes the range (http://planetmath.org/Function) of an operator .
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.
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for every .
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
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Let be a measure space. Consider the Hilbert space (http://planetmath.org/L2SpacesAreHilbertSpaces). We regard a function in (http://planetmath.org/LpSpace) as the multiplication operator given by
In this setting, the characteristic functions are projections in and we have a spectral measure given by
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Let be a Hilbert space, a normal operator and the spectrum of . For any measurable subset the operators , given by the Borel functional calculus, are projections in . Moreover, we have a spectral measure given by:
3 Equivalent Definition
The following result provides a very useful equivalent definition of a spectral measure.
Theorem 1 - A function whose values are projections is a spectral measure in if and only if and for every the function given by
is a complex measure in .
4 Integration against spectral measures
Let be a bounded (http://planetmath.org/Bounded) measurable function and a spectral measure in . We are interested to give meaning to the integral
Since we are dealing with “measures” whose values are linear operators 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 by simple functions 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 with the following property: for any given and for every measurable partition of that satisfies for all , we have
for any choice of points .
We can then define as the unique operator described by Theorem 2.
The other approach to define this integral is by specifying an appropriate bounded sesquilinear form. Recall that from Riesz representation theorem (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 we can define a complex measure by
whose total variation is estimated by .
Then, by the Riesz representation theorem (http://planetmath.org/RieszRepresentationTheoremOfBoundedSesquilinearForms), there exists a unique operator such that
(1) |
We can then define as this operator . Of course, the two definitions are equivalent. We summarize this in the following result
Theorem 3 - Given a spectral measure and a bounded Borel function , an operator that satisfies condition (1) also satisfies the conditions of Theorem 2. Therefore, both definitions of the integral of with respect to coincide and we have that:
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can be arbitrarilly approximated in norm by operators of the form .
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 theorem.
References
- 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 Analysis, 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 |