Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$.
Show that one of the two following assertions is true: (i) $\operatorname{ker}(T) \subset F^+$, (ii) $\operatorname{ker}(T) \subset F^-$.