Let T PR PR be an injective operator such that deg Tp deg p

Let T : P(R) P(R) be an injective operator such that deg T(p) deg p for every nonzero polynomial p P(R).
(i) Prove that T is also surjective.

(ii) Prove that deg T(p) = deg p for every nonzero p P(R).

Solution

Restrict T to the finite-dimensional subspace of polynomials with degree d. This restriction is a bijection, since T is injective. Suppose some polynomial of degree d is mapped to a polynomial of degree <d, then T cannot be surjective, since there is a polynomial of degree <d also mapped to that polynomial. So T maps polynomials of degree dd to polynomials of degree dd and is surjective