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

## Proof

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $II.3.7$