$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, and $U$ is the endomorphism of $\mathcal{P}$ defined by $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$. Let $\lambda$ be an eigenvalue of $U$ and $P$ an eigenvector associated with it. VIII.F.1) Show that $P$ is a solution of a simple linear differential equation that we will specify. VIII.F.2) What is the relationship between $\lambda$ and the degree of $P$?
$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, and $U$ is the endomorphism of $\mathcal{P}$ defined by $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$.
Let $\lambda$ be an eigenvalue of $U$ and $P$ an eigenvector associated with it.
VIII.F.1) Show that $P$ is a solution of a simple linear differential equation that we will specify.
VIII.F.2) What is the relationship between $\lambda$ and the degree of $P$?