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