a) is a projection in for every .
c) , where denotes the identity operator in .
d) If and are disjoint subsets of , then and are orthogonal.
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 .
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.
In this setting, the characteristic functions are projections in and 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
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
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 we notice that the function defined by
is a sesquilinear form.
Then, by the Riesz representation theorem (http://planetmath.org/RieszRepresentationTheoremOfBoundedSesquilinearForms), there exists a unique operator such that
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:
can be arbitrarilly approximated in norm by operators of the form .
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.
|Date of creation||2013-03-22 17:32:06|
|Last modified on||2013-03-22 17:32:06|
|Last modified by||asteroid (17536)|
|Synonym||projection valued measure|
|Defines||integration against spectral measures|