$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$.
We denote by $D$ the differentiation endomorphism and $U$ the endomorphism of $\mathcal{P}$ defined by
$$U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right).$$
Verify that $U$ is an endomorphism of $\mathcal{P}$.