Characterization of Reducing Subspaces

Theorem
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 $P$ the orthogonal projection on $M$.

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

Then the following four statements are equivalent:


 * $(1): \qquad M$ is a reducing subspace for $A$
 * $(2): \qquad PA = AP$
 * $(3): \qquad X = Y = 0$
 * $(4): \qquad M$ is an invariant subspace for both $A$ and its adjoint $A^*$