Lemma 6

Let $\pi$ and $\varrho$ be unitary representations of a topological group G in Hilbert spaces E and F respectively. Let D be a dense subspace of E which is stable under $\pi$. Let $u$ be a closed partial operator from E into F with domain D such that $u\circ \pi (g) =\varrho (g)\circ u$ for every $g\in G$. Then the domain of $u^*$ is stable under $\varrho$ and one has the relation $u^*\circ \varrho (g) =\pi (g)\circ u^*$ for every $g\in G$.

Let $g\in G$ and $x\in$ dom($u^*$). For every $y\in$ dom($u$), one has

since $\varrho (g)^*=\varrho (g)^{-1}=\varrho (g^{-1})$. This proves that $\varrho (g)x\in$ dom($u^*$) and that $u^*(\varrho (g)x) =\pi (g)(u^*(x))$. In particular, the domain of $u^*\circ \varrho (g)$ contains the domain of $u^*$, and $\pi (g)\circ u^*\subset u^*\circ \varrho (g)$.

But moreover, if $x\in$ dom($u^*\circ \varrho (g)$), then $x=\varrho (g^{-1})(\varrho (g)x)$ belongs to dom($u^*$) by what precedes applied to $g^{-1}$. It follows that $u^*\circ \varrho (g) =\pi (g)\circ u^*$.