= Spectral projection gives a reducing subspace
For a bounded <normal operator> $T$ with <spectral measure of a normal operator> $P$, any projection $P(E)$ commutes with $T$ and $T^*$ by the <Borel functional calculus for a normal operator>. Its range is a closed reducing subspace. If the spectrum has two points, choose disjoint nonempty relative open sets around them. <Full support of a faithful spectral measure> makes both projections nonzero and their product zero, so either range is nonzero and proper, in particular an <invariant subspace>.
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