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) $\forall v \in E, S(u,v) = 0$, where $S(v,w) = (v \mid T(w)) + (T(v) \mid w)$. We assume here that $\operatorname{ker}(T) \subset F^+$. (a) Show that $\forall z \in F^-, T^{2m}(z) = 0_E$. (b) Show that $\operatorname{Im}(T)^\perp \subset F^+$ and that $\operatorname{Im}(T^2)^\perp \cap \operatorname{Im}(T) \subset F^-$. (c) Let $z \in \operatorname{Im}(T)^\perp$ with $z \neq 0_E$. Show that $T(z) \in G^\perp$ and that $T(z) \neq 0_E$. (d) Let $z \in \operatorname{Im}(T^2)^\perp \cap \operatorname{Im}(T)$ with $z \neq 0_E$. Show that $T(z) \in G^\perp$ and that $T(z) \neq 0_E$.
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) $\forall v \in E, S(u,v) = 0$, where $S(v,w) = (v \mid T(w)) + (T(v) \mid w)$.
We assume here that $\operatorname{ker}(T) \subset F^+$.\\
(a) Show that $\forall z \in F^-, T^{2m}(z) = 0_E$.\\
(b) Show that $\operatorname{Im}(T)^\perp \subset F^+$ and that $\operatorname{Im}(T^2)^\perp \cap \operatorname{Im}(T) \subset F^-$.\\
(c) Let $z \in \operatorname{Im}(T)^\perp$ with $z \neq 0_E$. Show that $T(z) \in G^\perp$ and that $T(z) \neq 0_E$.\\
(d) Let $z \in \operatorname{Im}(T^2)^\perp \cap \operatorname{Im}(T)$ with $z \neq 0_E$. Show that $T(z) \in G^\perp$ and that $T(z) \neq 0_E$.