Corollary 1

Let $\pi$ be an irreducible unitary representation of a topological group G in a Hilbert space E. Let $u$ be a closed partial operator on E. Suppose that $u$ has dense domain, that dom($u$) is stable under $\pi$ and that $u\circ \pi (g) =\pi (g)\circ u$ for all $g\in G$. Then dom($u$) $= E$ and $u$ is a homothety.

The partial operator $u^*\circ u$ is a positive self-adjoint partial operator on E (Prop. 12 of IV, p. 241); the same is true of $v= 1_E+u^*\circ u$, and the latter is injective since $-1\notin$ Sp($u^*\circ u$) (Prop. 17 of IV, p. 248). We have $u^*\circ \pi (g) =\pi (g)\circ u^*$ for all $g\in G$ (Lemma 6 of V, p. 381), whence $v\circ \pi (g) =\pi (g)\circ v$ for all $g\in G$. Since $v$ is injective, it follows that $v^{-1}\circ \pi (g) =\pi (g)\circ v^{-1}$ for all $g\in G$. But $v^{-1}$ belongs to $\mathscr{L}$ (E), hence by Prop. 6 there exists $\lambda \in \mathbf{C}$ such that $v^{-1}=\lambda 1_E$. Necessarily $\lambda \not = 0$, which implies that E = Im($v^{-1}$) $=$ dom($v$)$\subset$ dom($u$), whence dom($u$) $= E$. Since $u$ is closed, we have $u\in \mathscr{L}(E)$ (TVS, I, p. 19, Cor. 5), hence $u\in$ Hom$_G(\pi , \pi )$ and $u$ is a homothety by Prop. 6.