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): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$, and (H4): $T \circ M + M \circ T = 0_{\mathcal{L}(E)}$.
Let $k \in \mathbb{N}$.\\
(a) Show that $M \circ T^k = (-1)^k T^k \circ M$.\\
(b) Deduce that $\operatorname{Im}(T^k)$ and $\operatorname{ker}(T^k)$ are stable under $M$.