Essential spectrum of a projection

Question

For any $T\in B(H)$, the essential spectrum $\sigma_e(T)$ of $T$ is a subset of the spectrum $\sigma(T)$ of $T$; namely, the $\lambda$ such that $\lambda-T$ is not Fredholm.

If $P$ is a projection, we have $\sigma_e(P)\subset \sigma(P)=\{0,1\}$. My question is whether $\sigma_e(P)$ can be $\{0\}$,$\{1\}$,$\{0,1\}$.

Can we find some concrete examples to show that the above three cases can happen?

Answer 1

Yes!

1. a projection onto a finite dimensional space,
2. a projection onto a finite co-dimensional space,
3. a projection not of the above types.