Definition:Reducing Subspace

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Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be a bounded linear operator.

Let $M$ be a closed linear subspace of $H$; denote by $M^\perp$ its orthocomplement.

Then $M$ is said to be a reducing subspace for $A$ iff both $M$ and $M^\perp$ are invariant subspaces for $A$.

That is, if $AM \subseteq M$ and $A M^\perp \subseteq M^\perp$.