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 $A^*$

Proof

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Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.3.7$