Disconnected spectrum gives a reducing subspace
= Disconnected spectrum gives a reducing subspace
{title2=$P=\mathbf1_{K_1}(T)$}
For a <normal operator> with disconnected <spectrum> $K=K_1\sqcup K_2$, the <indicator function> of a nonempty proper clopen component union is continuous on $K$. The <continuous functional calculus> makes it a nonzero proper self-adjoint <linear projection> commuting with $T,T^*$. Its range is therefore a nontrivial closed reducing subspace. No <eigenvector> is needed.