\begin{definition}[Definition:Reducing Subspace]
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 $A M^\perp \subseteq M^\perp$.
\end{definition}