Definition:Reducing Subspace

Definition
Let $H$ be a Hilbert space.

Let $A \in \map B H$ 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 $A$ both $M$ and $M^\perp$ are invariant subspaces for $A$.

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