| 1. | The continuous, or just the holomorphic functional calculus suffices.
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| 2. | For the continuous functional calculus, the key ingredients are the following:
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| 3. | This property will be used in subsequent arguments for the functional calculus.
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| 4. | By property 3 of the functional calculus, the operator
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| 5. | This definition can be generalized to include continuous functions using continuous functional calculus.
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| 6. | The well-definedness of functional calculus now follows as an easy consequence.
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| 7. | Thus a more general functional calculus is needed.
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| 8. | First pass from polynomial to continuous functional calculus by using the Stone-Weierstrass theorem.
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| 9. | Is positive, where the continuous functional calculus is used to define the square root.
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| 10. | Property 2 and the continuous functional calculus ensure that ? preserves the *-operation.
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