proof of Borel functional calculus
In this entry we give a proof of the main result about the Borel functional calculus ( 1 on the parent entry). We will restate here the result for convenience. Please, check the parent entry for the details on notation.
Theorem - Let be a normal operator in and the unital *-homomorphism corresponding to the continuous functional calculus for . Then, extends uniquely to a *-homomorphism that is continuous from the -topology to the weak operator topology. Moreover, each operator lies in strong operator (http://planetmath.org/OperatorTopologies) closure of the unital *-algebra generated by .
Proof : First we shall prove the existence of the extension (http://planetmath.org/ExtensionOfAFunction) with the described continuity property, then we shall prove its uniqueness and for last we shall prove the last assertion of the theorem about the image of . For simplicity the proofs of some auxiliary results are given at the end of the entry
For each pair of vectors consider the linear functional given by
This linear functional is bounded with norm at most because
where the last equality comes from the fact that is a *-isomorphism between -algebras (http://planetmath.org/CAlgebras), therefore having norm 1 (see this entry (http://planetmath.org/HomomorphismsOfCAlgebrasAreContinuous)).
a) , for all .
b) , for all .
d) , for all .
Moreover, this sesquilinear form is bounded (http://planetmath.org/BoundedSesquilinearForm) with norm at most . By the Riesz lemma on bounded sesquilinear forms, there is a unique operator such that
We will now see that the mapping , such that , has the desired properties stated in the theorem.
First, it is clear that is linear. Also clear is the fact that coincides with in , because of equality (1) and the uniqueness part of Riesz lemma on sesquilinear forms. Now, for any real valued function we have that
which means that , i.e. is self-adjoint. Decomposing an arbitrary function in its real and imaginary parts we see that .
We now show that is multiplicative, i.e. for all . For that we need an additional property of the measures , whoose proof is also at the end of the entry:
e) , for all .
Given we have, for every ,
and therefore . Thus, is a *-homomorphism that extends .
0.0.2 Continuity Property
We now prove that the above defined is continuous from the -topology to the weak operator topology.
Let be a net of functions in that converge in the -topology to a function . This means that for all Radon measures in we have .
Now for all we have
Hence, converges to in the weak operator topology.
Let be another *-homomorphism that extends and is continuous from the -topology to the weak operator topology. For any measurable subset consider the set
We give this set the partial order such that whenever and . For any pair there is a continuous function such that takes values on the interval , and (see this entry (http://planetmath.org/ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces)).
We claim that converges to in the -topology. In fact, given a complex Radon measure in , there is for every a pair such that . Of course, for all pairs such that we also have . Hence, we have
We conclude that converges to in the -topology.
Since is continuous from the -topology to the weak operator topology we must have
But since and coincide with on we also have
0.0.4 Image of
Let be the unital *-algebra generated by . We now prove that for any , the operator lies in the strong operator closure of , i.e. lies in the von Neumann algebra generated by . For that it is enough to prove that is in the double commutant of .
Recall from the continuous functional calculus that is in the norm closure of , and hence in , for every .
We have seen above that for each characteristic function there is a net of functions in such that in the weak operator topology. Given an element in the commutant of we have
The first term converges to , whereas the second to . Thus, , and therefore .
Since every function can be uniformly approximated by simple functions, it follows that .
0.0.5 Auxiliary Results
In this section we prove the properties of the measures stated and used above.
a) For all functions we have . Hence,
Since this holds for every , the uniqueness part of tells us that
b) The proof is similar to a).
c) For every we have
Hence we conclude that .
d) For every we have
e) For all we have
|Title||proof of Borel functional calculus|
|Date of creation||2013-03-22 18:50:26|
|Last modified on||2013-03-22 18:50:26|
|Last modified by||asteroid (17536)|