Proposition 3

Let $U$ be a composition operator of order $p$; then $U(f) = 0$ for every polynomial $f$ of degree $< p$; for every polynomial $f \neq 0$ of degree $n \geq p$, $U(f)$ is a polynomial $\neq 0$ of degree $n - p$.

This is an immediate consequence of formula (2) of VI, p. 271 and of the definition of the order of $U$.

It is clear that every operator $U$ of order $p$ can be written in a unique way as $U = D^p V = V D^p$, where $V$ is an operator of order 0 (and so invertible).