Exercise 13, § 2

Let $\pi$ be an irreducible unitary representation of G in a complex Hilbert space E. Let $\varrho$ be a unitary representation of G in a complex Hilbert space F. Suppose that there exists a closed partial operator with dense domain $u$ from E into F whose domain is a G-stable subspace of E and which satisfies $u\circ \pi (g) =\varrho (g)\circ u$ for every $g\in G$.

The partial operator $u$ belongs to $\mathscr{L}(E; F)$, and there exists $\lambda >0$ such that $\lambda u$ is isometric.