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Invariant subspace for normal operators

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Feb 15, 2014
I have proved the spectral theorem for a normal operator T on an infinite dimensional hilbert space, and am now trying to deduce that T has non-trivial invariant susbspaces.

Case 1: If the spectrum of T consists of a single point: My book says that if this is the case then the set of continuous functions of the spectrum is isomorphic to the complex numbers. Don't know how they got that

case 2: If the spectrum of T has more than one point, then my book says that are disjoint borel subsets whose union is the whole spectrum but again not sure how they got that.