This defines the functional calculus for " bounded " functions applied to possibly " unbounded " self-adjoint operators.
32.
The idea of a functional calculus is to create a " principled " approach to this kind of overloading of the notation.
33.
In the definition of the functional calculus, " f " is assumed to be holomorphic in an open neighborhood of ?.
34.
The holomorphic functional calculus is similar in that the resolvent mapping plays a crucial role in obtaining properties one requires from a nice functional calculus.
35.
The holomorphic functional calculus is similar in that the resolvent mapping plays a crucial role in obtaining properties one requires from a nice functional calculus.
36.
The Riesz-Markov theorem then allows us to pass from integration on continuous functions to spectral measures, and this is the Borel functional calculus.
37.
V * is the unique positive square root of " M * M ", as given by the Borel functional calculus for self adjoint operators.
38.
What has not been shown is that the definition of the functional calculus is unambiguous, i . e . does not depend on the choice of ?.
39.
In slightly more abstract language, the holomorphic functional calculus can be extended to any element of a Banach algebra, using essentially the same arguments as above.
40.
For technical reasons, one needs to consider separately the positive and negative parts of " A " defined by the Borel functional calculus for unbounded operators.
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