= Disconnected spectrum yields a nontrivial invariant subspace
{title2=$P^2=P,\quad PT=TP,\quad0\neq P\neq I$}
For a bounded operator on a complex <Banach space>, partition a disconnected spectrum into two nonempty compact pieces. The <holomorphic function> equal to one near the first piece and zero near the second defines a <Riesz projection>. The <holomorphic spectral mapping theorem> makes it neither zero nor identity. Its closed range is a proper nonzero invariant <vector subspace>. This need not be an orthogonal projection, and the argument does not require a normal operator.
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