Remark

Since K is an infinite field the operator U on K[X] is a composition operator if and only if for every polynomial $f \in K[X]$ and every element $\alpha \in K$ one has, putting $g = U(f)$, that $g(X + \alpha) = U(f(X + \alpha))$. (Alg., IV. 16, cor.).

It is clear that every linear combination of composition operators, with coefficients in K, is a composition operator; the same is true for the composition of two composition operators. In other words, the composition operators form a subalgebra $\Gamma$ of the algebra of endomorphisms of the vector space K[X].