Strongly continuous unitary representations, invariant linear subspaces and intertwiners

Let $G$ be a topological group and $H$ a complex Hilbert space. A unitary representation of $G$ on $H$ is a group homomorphism $\pi :G\to U(H)$, where $U(H)$ is the group of bijective complex-linear isometries of $H$, such that the orbit map $g\mapsto \pi (g)v$ is norm-continuous for every $v\in H$. This condition is strong continuity. A closed linear subspace $M\subseteq H$ is invariant if $\pi (g)M=M$ for every $g\in G$. The representation is irreducible if $H\mathrm{\ne }\{0\}$ and its only closed invariant linear subspaces are $\{0\}$ and $H$.

For unitary representations $\pi$ on $H$ and $\rho$ on $K$, a bounded intertwiner is a bounded linear operator $T:H\to K$ satisfying $T\pi (g)=\rho (g)T(g\in G)\mathrm{.}$ The representations are unitarily equivalent if there is a unitary intertwiner between them. The commutant of $\pi$ is $\pi (G{)}^{\mathrm{\prime }}=\{T\in \mathcal{B}(H):T\pi (g)=\pi (g)T\text{ for every }g\in G\}\mathrm{.}$