Let $T \in \mathcal{L}(E)$ be a polynomially compact operator, i.e., there exists a polynomial $p$ such that $p(T)$ is a compact operator. Suppose $p(1) \neq 0$. I want to show that $N(I-T) = \{0\} \Leftrightarrow R(I-T) = E$ and that $\dim N(I-T) = \dim N(I-T^*)$. I think I can show that $N(I-T) = \{0\} \Rightarrow R(I-T) = E$, but the other implications are eluding me.
I currently know that $I-T$ is Fredholm, that $N(p(I) - p(T)) \supseteq N(I-T)$, and $R(p(I) - p(T)) \subseteq R(I-T)$.
Where do I go from here?