Characterization of Invariant Subspaces

Theorem
Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be a bounded linear operator.

Let $\begin{pmatrix} W & X \\ Y & Z \end{pmatrix}$ be the matrix notation for $A$ with respect to $M$.

Let $M$ be a closed linear subspace of $H$; denote by $P$ the orthogonal projection on $M$.

Then the following three statements are equivalent:


 * $(1): \qquad M$ is an invariant subspace for $A$
 * $(2): \qquad PAP = AP$
 * $(3): \qquad Y = 0$