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With stronger assumptions, when T is a normal operator acting on a Hilbert space, the domain of the functional calculus can be broadened. When comparing the two results, a rough analogy can be made with the relationship between the spectral theorem for normal matrices and the Jordan canonical form. When T is a normal operator, a continuous functional calculus can be obtained, that is, one can evaluate f(T) with f being a continuous function defined on σ(T). Using the machinery of measure theory, this can be extended to functions which are only measurable (see Borel functional calculus). In that context, if E ⊂ σ(T) is a Borel set and E(x) is the characteristic function of E, the projection operator E(T) is a refinement of ei(T) discussed above.
The Borel functional calculus extends to unbounded self-adjoint operators on a Hilbert space.
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. Similarly, the continuous functional calculus holds for normal elements in any C*-algebra and the measurable functional calculus for normal elements in any von Neumann algebra.
Read more about this topic: Holomorphic Functional Calculus
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