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 $A$ if and only if both $M$ and $M^\perp$ are invariant subspaces for $A$.

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

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.3.5$