We keep all the notations from Part I and we assume that hypotheses (H1), (H2), (H3), (H4) and (H5) are all satisfied. Let $(w_1, w_2)$ be a characterizing pair of $G$ (satisfying properties (A), (B) and (C) of question 12). For any $\lambda \in \mathbb{R}$, we consider the following problem, denoted $\mathcal{P}_\lambda$: $$\text{Find } u \in G \text{ such that: } \forall v \in G, (u \mid v) - \lambda (T(u) \mid T(v)) = 0$$ and we denote by $H_\lambda$ the set of solutions $u$ of this problem. (a) Show that if $(\mathcal{P}_\lambda)$ admits a solution $u \neq 0_E$, then necessarily $\lambda > 0$. (b) Let $u \in G$. Show that $u$ is a solution of $(\mathcal{P}_\lambda)$ if and only if $$\left(\operatorname{Id}_E + \lambda T^2\right)(u) \in G^\perp$$ Deduce that there exist two real numbers $\alpha$ and $\beta$ such that: $$u = \alpha \left(\operatorname{Id}_E + \lambda T^2\right)^{-1} T(w_1) + \beta \left(\operatorname{Id}_E + \lambda T^2\right)^{-1} T(w_2)$$ (c) Show that the problem $(\mathcal{P}_\lambda)$ admits a non-zero solution if and only if $$Q_1(\lambda) \cdot Q_2(\lambda) = 0$$ where for $i \in \{1,2\}$, $Q_i$ is the polynomial $$Q_i(X) = \sum_{k=0}^{m-1} (-1)^k \left(T^{2k+1}(w_i) \mid T(w_i)\right) X^k$$ (d) Suppose that $\lambda$ is a root of the product polynomial $Q_1 Q_2$. Show that $\operatorname{dim}(H_\lambda) = 2$ if $\lambda$ is a common root of $Q_1$ and $Q_2$, and $\operatorname{dim}(H_\lambda) = 1$ otherwise. (e) Show that $$\forall i \in \{1,2\}, Q_i(X) = \sum_{k=0}^{m-1} (-1)^k S\left(w_i, T^{2k+1}(w_i)\right) X^k$$
We keep all the notations from Part I and we assume that hypotheses (H1), (H2), (H3), (H4) and (H5) are all satisfied. Let $(w_1, w_2)$ be a characterizing pair of $G$ (satisfying properties (A), (B) and (C) of question 12). For any $\lambda \in \mathbb{R}$, we consider the following problem, denoted $\mathcal{P}_\lambda$:
$$\text{Find } u \in G \text{ such that: } \forall v \in G, (u \mid v) - \lambda (T(u) \mid T(v)) = 0$$
and we denote by $H_\lambda$ the set of solutions $u$ of this problem.
(a) Show that if $(\mathcal{P}_\lambda)$ admits a solution $u \neq 0_E$, then necessarily $\lambda > 0$.\\
(b) Let $u \in G$. Show that $u$ is a solution of $(\mathcal{P}_\lambda)$ if and only if
$$\left(\operatorname{Id}_E + \lambda T^2\right)(u) \in G^\perp$$
Deduce that there exist two real numbers $\alpha$ and $\beta$ such that:
$$u = \alpha \left(\operatorname{Id}_E + \lambda T^2\right)^{-1} T(w_1) + \beta \left(\operatorname{Id}_E + \lambda T^2\right)^{-1} T(w_2)$$
(c) Show that the problem $(\mathcal{P}_\lambda)$ admits a non-zero solution if and only if
$$Q_1(\lambda) \cdot Q_2(\lambda) = 0$$
where for $i \in \{1,2\}$, $Q_i$ is the polynomial
$$Q_i(X) = \sum_{k=0}^{m-1} (-1)^k \left(T^{2k+1}(w_i) \mid T(w_i)\right) X^k$$
(d) Suppose that $\lambda$ is a root of the product polynomial $Q_1 Q_2$. Show that $\operatorname{dim}(H_\lambda) = 2$ if $\lambda$ is a common root of $Q_1$ and $Q_2$, and $\operatorname{dim}(H_\lambda) = 1$ otherwise.\\
(e) Show that
$$\forall i \in \{1,2\}, Q_i(X) = \sum_{k=0}^{m-1} (-1)^k S\left(w_i, T^{2k+1}(w_i)\right) X^k$$