Let $H$ be a real or complex [Hilbert space](/page/Hilbert%20Space), let $T\in\mathcal{L}(H)$, and let $M\subset H$ be a closed linear subspace. Let $P_M\in\mathcal{L}(H)$ denote the [orthogonal projection](/theorems/437) onto $M$. The following are equivalent:
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(i) $M$ reduces $T$, meaning $T(M)\subset M$ and $T(M^\perp)\subset M^\perp$.
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(ii) $M$ is invariant under both $T$ and $T^*$, meaning $T(M)\subset M$ and $T^*(M)\subset M$.