Definition:Invariant 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 $H$.

Then $M$ is said to be an invariant subspace for $A$ $h \in M \implies A h \in M$.

That is, $AM \subseteq M$.

Also see

 * Definition:Reducing Subspace