Definition 32.1.6 (Subrepresentation).label Let $G$ be a locally compact group, $(H, \pi)$ be a unitary representation of $G$, and $M \subset H$ be an invariant subspace, then the mapping
\[\pi_{M}: G \to U(B(M)) \quad x \mapsto \pi(x)|_{M}\]
is a unitary subrepresentation of $G$.
If $\pi$ admits a non-trivial invariant subspace, then $\pi$ is reducible. Otherwise, it is irreducible.
Post a Comment