Proposition 8

Let G be a topological group. Let $\varrho$ be a unitary representation of G in a complex Hilbert space E.

a) Let $\pi_1$ and $\pi_2$ be non-isomorphic irreducible unitary representations of G. The subspaces $M_{\pi_1}(\varrho )$ and $M_{\pi_2}(\varrho )$ are orthogonal;

b) Let $\varrho '$ be a unitary representation of G and $u$ a G-morphism of $\varrho$ into $\varrho '$. For every irreducible unitary representation $\pi$ of G, one has $u(M_{\pi}(\varrho ))\subset M_{\pi}(\varrho ')$.

Let $E_1$ and $E_2$ be subspaces of E defining subrepresentations isomorphic to $\pi_1$ and $\pi_2$ respectively. The orthogonal projector of E with image $E_2$ defines, by passing to the subspaces, an element of Hom$_G(\pi_1, \pi_2)$, which is zero (Cor. 2 of V, p. 387), and this proves that $E_2$ is orthogonal to $E_1$. This implies assertion a).

As for assertion b), note that $M_{\pi}(\varrho )$ is, by definition, the closure in E of the space generated by the elements $x\in E$ belonging to a closed subspace $F\subset E$ stable under $\varrho$ such that the subrepresentation $\varrho_F$ of $\varrho$ in F is isomorphic to $\pi$. It is therefore enough to prove that, in this case, one has $u(x)\in M_{\pi}(\varrho ')$. Let $H =u(F)$; it is a closed subspace of the space of $\varrho '$ (Cor. 5 of V, p. 388) stable under $\varrho '$, and $u$ induces, by passing to the subspaces, a surjective G-morphism of F onto H. If H is non-zero, then this G-morphism is an isomorphism by Corollary 2 of V, p. 387. The representation of G on H is therefore isomorphic to $\pi$, whence it follows that $u(x)$ belongs to $M_{\pi}(\varrho ')$.

For every vector space H and every family $(H_i)_{i\in I}$ of subspaces of H, one says that the spaces $(H_i)_{i\in I}$ are in direct sum if the family $(H_i)_{i\in I}$ satisfies the equivalent conditions of Prop. 11 of A, II, p. 18.