Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$. Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$. We now say that a pair $(w_1, w_2) \in E \times E$ is a characterizing pair of $G$ if $w_1$ and $w_2$ satisfy the three properties: (A) $w_1 \in F^+$, $T(w_1) \in G^\perp$ and $T(w_1) \neq 0_E$, (B) $w_2 \in F^-$, $T(w_2) \in G^\perp$ and $T(w_2) \neq 0_E$, (C) $w_i \in \operatorname{Im}(T^2)^\perp$ for $i = 1$ and $i = 2$. Deduce from the previous questions the existence of a characterizing pair of $G$.
Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$. Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$.
We now say that a pair $(w_1, w_2) \in E \times E$ is a characterizing pair of $G$ if $w_1$ and $w_2$ satisfy the three properties:\\
(A) $w_1 \in F^+$, $T(w_1) \in G^\perp$ and $T(w_1) \neq 0_E$,\\
(B) $w_2 \in F^-$, $T(w_2) \in G^\perp$ and $T(w_2) \neq 0_E$,\\
(C) $w_i \in \operatorname{Im}(T^2)^\perp$ for $i = 1$ and $i = 2$.
Deduce from the previous questions the existence of a characterizing pair of $G$.