Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$. Show that for any real number $\alpha$, the endomorphism $\operatorname{Id}_E + \alpha T^2$ is bijective and that $$\left(\operatorname{Id}_E + \alpha T^2\right)^{-1} = \sum_{k=0}^{m} (-1)^k \alpha^k T^{2k}$$ where $\left(\operatorname{Id}_E + \alpha T^2\right)^{-1}$ denotes the inverse endomorphism of $\operatorname{Id}_E + \alpha T^2$.
Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.
Show that for any real number $\alpha$, the endomorphism $\operatorname{Id}_E + \alpha T^2$ is bijective and that
$$\left(\operatorname{Id}_E + \alpha T^2\right)^{-1} = \sum_{k=0}^{m} (-1)^k \alpha^k T^{2k}$$
where $\left(\operatorname{Id}_E + \alpha T^2\right)^{-1}$ denotes the inverse endomorphism of $\operatorname{Id}_E + \alpha T^2$.