Clarification: is P(R) the set of all polynomials on the reals? And is P2(R) the set of all quadratics? If so, you tell me: is multiplication by x going to leave a quadratic a quadratic?

I should also point out that "T-invariant" might mean different things depending on definition. Does it mean that T is a bijection from a space onto itself? Or is it allowed to map a subspace to a subset of the subspace (i.e., can it fail to be onto?)