Let $X$ be a Banach space and $C$ be a compact operator on $X$.
Then for a fixed nonzero eigenvalue $\lambda$ of $C$, I know that the resolvent $(\xi-C)^{-1}$ has a Laurent expansion near $\lambda$ with a pole at $\lambda$.
Let $(\xi-C)^{-1}=\sum_{-m}^{\infty}A_j(\xi-\lambda)^j$. Then I showed that $A_{-1}$ is actually a projection, i.e $A_{-1}^2=A_{-1}$.
However I have trouble figuring out what the range of $A_{-1}$ is. I know that for some natural number $l$, the null space $N_l$ of $(\lambda-C)^m$ is stable. That is, $N_l=N_i$ for all $i \geq l$.
I have to show taht the range of $A_{-1}$ is this $N_l$. However I cannot find a way to do this. Actually this is an exercise from Lax Functional Analysis p.240~241. Could anyone please help me?