= Reducing subspace of a Hilbert-space operator
{title2=$TM\subseteq M,\quad T^*M\subseteq M$}
= Reducing subspace
{synonym}
A <closed> subspace of a <Hilbert space> reduces a <bounded operator> when both the operator and its adjoint leave the subspace invariant. Equivalently, the <orthogonal> projection onto it commutes with the operator: the adjoint invariance makes its <orthogonal complement> invariant too. For a star-closed algebra of operators, the <closed> span of the orbit of any <vector> is reducing for the whole algebra. This enables an <orthogonal> direct-sum decomposition into cyclic actions when constructing <Borel functional calculus for a normal operator>.
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