Definition:Invariant Subspace

Definition

Let $H$ be a Hilbert space.

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

Let $M$ be a closed linear subspace of $H$.

Then $M$ is said to be an invariant subspace for $A$ if and only if $h \in M \implies A h \in M$.

That is, if and only if $AM \subseteq M$.

Also see

• Definition:Reducing Subspace

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text{II}.3.5$