I am asked to compute the set of all $\lambda$'s such that $T - \lambda I$ is not a Fredholm operator (where $I$ is the identity map), if $T$ is the left shift operator.
The underlying space is $\ell^p$ (we can assume $p$ if finite) and the left shift operator $T_l : \ell^p \to \ell^p$ sends $(x_1,x_2,x_3,...) \to (x_2,x_3,...)$.
So far I have been able to show the Kernel is always finite dimensional, independently of the choice of $\lambda \in \mathbb{C}$ and that the map is surjective (with trivial cokernel) for any $\lambda$ such that $|\lambda| < 1$. So in conclussion I could prove so far that the $\lambda$ s.t. $|\lambda| < 1$ is not in the essential spectrum.
From this point on I am really stucked. Any hints are very welcome.