| 21. | These operators always have a canonically defined Friedrichs extension and for these operators we can define a canonical functional calculus.
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| 22. | In the finite-dimensional case, the polynomial functional calculus yields quite a bit of information about the operator.
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| 23. | The functional calculus can be defined in exactly the same way for an element in " A ".
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| 24. | Some of these properties can be established by using the continuous functional calculus or by reduction to commutative C *-algebras.
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| 25. | By the functional calculus, this C *-algebra is the continuous functions on the unit circle in the complex plane.
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| 26. | Using the bounded functional calculus, one can prove part of the Stone's theorem on one-parameter unitary groups:
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| 27. | The idea is to first establish the continuous functional calculus then pass to measurable functions via the Riesz-Markov representation theorem.
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| 28. | Applying the spectral theorem, or Borel functional calculus for infinite dimensional systems, we see that it generalizes the classical entropy.
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| 29. | Using the machinery of measure theory, this can be extended to functions which are only measurable ( see Borel functional calculus ).
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| 30. | In more modern treatments however, this representation is usually avoided, since most technical problems can be dealt with by the functional calculus.
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