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 $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.
Let $k \in \{1, 2, \ldots, 2m+1\}$ and $z \in \operatorname{Im}(T^k)^\perp \cap \operatorname{Im}(T^{k-1})$ such that $z \neq 0_E$. After justifying the existence of such a vector $z$, show that $T^{2m+1-k}(z) \neq 0_E$.