Definition:Invariant Subspace

Definition
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 $H$.

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

That is, if $AM \subseteq M$.

Also see

 * Reducing Subspace