Characterization of Invariant Subspaces

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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 $A$.

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 $A$
$(2): \qquad PAP = AP$
$(3): \qquad Y = 0$


Proof


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