So I'm stuck at the following result, about compact operators on Hilbert spaces (which I think it's called Fredholm's theorem) from Stein. It's exercise 29 from Chapter 4.
Let $T$ be a compact operator on a Hilbert space $\mathcal{H}$, and assume $\lambda \neq 0$.
a) Show that the range of $\lambda I - T$ is closed.
b) Show that this is not true for $\lambda = 0$.
c) Show that the range of $\lambda I - T$ is all of $\mathcal{H}$ if and only if the nullspace of $\overline{\lambda} I - T^{*}$ is trivial.
Sorry if it's already here on the forum! Thanks