The linear application $\widehat{\xi}$ and the endomorphism $\xi$ We denote by $\widehat{\xi} : \mathbb{C}[X^{\pm 1}] \rightarrow \mathcal{D}$ the linear application that to a Laurent polynomial $F$ associates $$\widehat{\xi}(F) = \Pi(XF) \quad \text{and} \quad \xi = \widehat{\xi}_{\mathcal{D}}$$ that is the endomorphism of $\mathcal{D}$ induced by $\widehat{\xi}$. a) Let $F$ be an element of $\mathbb{C}[X^{\pm 1}]$. Prove that $\widehat{\xi}(\Pi(F)) = \widehat{\xi}(F)$. b) Let $P$ be a polynomial and let $F$ be an element of $\mathcal{D}$. Prove that $P(\xi)(F) = \Pi(PF)$.
\textbf{The linear application $\widehat{\xi}$ and the endomorphism $\xi$}\\
We denote by $\widehat{\xi} : \mathbb{C}[X^{\pm 1}] \rightarrow \mathcal{D}$ the linear application that to a Laurent polynomial $F$ associates
$$\widehat{\xi}(F) = \Pi(XF) \quad \text{and} \quad \xi = \widehat{\xi}_{\mathcal{D}}$$
that is the endomorphism of $\mathcal{D}$ induced by $\widehat{\xi}$.
a) Let $F$ be an element of $\mathbb{C}[X^{\pm 1}]$. Prove that $\widehat{\xi}(\Pi(F)) = \widehat{\xi}(F)$.
b) Let $P$ be a polynomial and let $F$ be an element of $\mathcal{D}$. Prove that $P(\xi)(F) = \Pi(PF)$.