The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ In this question, we propose to show that there does not exist a linear application $u : \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X]$ such that $u \circ u = \delta$. We suppose, by contradiction, that such an application $u$ exists. a) Show that $u$ and $\delta^2$ commute. b) Deduce that $\mathbb{R}_1[X]$ is stable under the application $u$. c) Show that there does not exist a matrix $A \in \mathcal{M}_2(\mathbb{R})$ such that $$A^2 = \left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right)$$ d) Conclude.
The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$:
$$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$
In this question, we propose to show that there does not exist a linear application $u : \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X]$ such that $u \circ u = \delta$. We suppose, by contradiction, that such an application $u$ exists.
a) Show that $u$ and $\delta^2$ commute.
b) Deduce that $\mathbb{R}_1[X]$ is stable under the application $u$.
c) Show that there does not exist a matrix $A \in \mathcal{M}_2(\mathbb{R})$ such that
$$A^2 = \left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right)$$
d) Conclude.