Characterization of Reducing Subspaces

Theorem
Let $\HH$ be a Hilbert space.

Let $A \in \map B \HH$ be a bounded linear operator.

Let $M$ be a closed linear subspace of $\HH$.

Let $P$ denote 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:


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