Corollary 4

Let $\varphi'$ be another analytic linear representation of $G$ on a complete normable space $E'$. Let $T \in \mathscr{L}(E, E')$. The following conditions are equivalent:

(i) $T\varphi(g) = \varphi'(g)T$ for all $g \in G$; (ii) $TL(\varphi)(a) = L(\varphi')(a)T$ for all $a \in L(G)$.

Let $\sigma$ be the linear representation of $G$ on $\mathscr{L}(E, E')$ derived from $\varphi$ and $\varphi'$ (§ 3, no. 11, Corollary 1 to Proposition 41). Condition (i) means that $T$ is invariant under $\sigma(G)$. Condition (ii) means that $T$ is annihilated by $L(\sigma)(L(G))$. It then suffices to apply Corollary 3.