Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. This space is equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying the following hypotheses: (H1) $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$. (H2) $M^2 = \operatorname{Id}_E$. (H3) $\forall (v,w) \in E^2, (M(v) \mid w) = (v \mid M(w))$. (H4) $T \circ M + M \circ T = 0_{\mathcal{L}(E)}$. We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$. For any vector $v \in E$, we set $$v^+ = v + M(v), \quad v^- = v - M(v)$$ (a) Show that $\forall v \in E, v^+ \in F^+$ and $v^- \in F^-$. (b) Show that $E = F^+ \oplus^\perp F^-$. (c) Show that $\forall v \in F^+, T(v) \in F^-$ and that $\forall v \in F^-, T(v) \in F^+$. Deduce that $F^+$ and $F^-$ are stable under $T^2$.
Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. This space is equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying the following hypotheses:\\
(H1) $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.\\
(H2) $M^2 = \operatorname{Id}_E$.\\
(H3) $\forall (v,w) \in E^2, (M(v) \mid w) = (v \mid M(w))$.\\
(H4) $T \circ M + M \circ T = 0_{\mathcal{L}(E)}$.\\
We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$.
For any vector $v \in E$, we set
$$v^+ = v + M(v), \quad v^- = v - M(v)$$
(a) Show that $\forall v \in E, v^+ \in F^+$ and $v^- \in F^-$.\\
(b) Show that $E = F^+ \oplus^\perp F^-$.\\
(c) Show that $\forall v \in F^+, T(v) \in F^-$ and that $\forall v \in F^-, T(v) \in F^+$.
Deduce that $F^+$ and $F^-$ are stable under $T^2$.