Proposition 4

Let $\varrho$ be a unitary representation of a topological group G in a Hilbert space E. A closed subspace F of E is a subrepresentation of $\varrho$ if and only if the orthogonal projector $p$ of E with image F is a G-morphism of $\varrho$ into $\varrho$.

Suppose that F is a subrepresentation of $\varrho$. Let $x\in E$ and write $x=p(x) +y$ where $y\in F^{\circ}$. For every $g$ in G, one has

and as $\varrho (g)(p(x))$ belongs to F and $\varrho (g)y$ to $F^{\circ}$ (Prop. 3), one has $p(\varrho (g)x) =\varrho (g)(p(x))$. Hence $p$ belongs to Hom$_G(\varrho , \varrho )$.

Conversely, if $p\in$ Hom$_G(\varrho , \varrho )$ then $1_E-p$ is a G-morphism, hence F = Ker(1$_E-p$) is a subrepresentation of $\varrho$.