We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product
$$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$
The two endomorphisms $T$ and $M$ of $E$ are defined by
$$\forall P \in \mathbb{R}_{2m}[X], \quad T(P) = P' \text{ and } M(P) = P^*$$
where $P^*(X) = P(-X)$.
Show that $T$ and $M$ satisfy hypotheses (H1), (H2), (H3) and (H4).