4. Borelian functional calculus

The above results are based on the continuous functional calculus and associated positive measures $l (h) = (h (T) x | x)$ where x is cyclic. At the price of abandoning the isometric character and, correlatively, the positivity of the measure, it is possible to extend the continuous calculus to bounded Borelian functions, which makes it possible to define projectors generalizing those of the holomorphic calculus.

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Borelian functional calculus

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Bibliography

(1) - ARVESON (W.) - A short course on spectral theory - Springer, Graduate texts in Mathematics 209 (2002).
(2) - ARVESON (W.) - An invitation to C* -algebras - Springer, Graduate texts in Mathematics 39 (1976).
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