Let $E$ be an infinite-dimensional complex Banach space. Let $\mathcal{L} (E)$ be the space of endomorphisms of $E$, endowed with the operator norm. Then $\mathcal{L} (E)$ is a unital Banach algebra for the usual operator composition. Let $\mathcal{K} (E)$ be the subspace of compact operators. It is well-know that:
$\mathcal{K} (E)$ is a non-trivial closed vector subspace of $\mathcal{L} (E)$;
$\mathcal{K} (E)$ is a bilateral ideal of $\mathcal{L} (E)$.
From the first property we deduce that $\mathcal{H} (E) := \mathcal{L} (E) / \mathcal{K} (E)$ is a Banach space for the norm $\| \tilde{T}\|_{\mathcal{H} (E)} = \inf \{\|T\|_{\mathcal{L} (E)}: \ \pi (T) = \tilde{T}\}$, where $\pi$ is the canonical projection. From the second property we deduce that the composition goes down to the quotient, and gives it a structure of unital Banach algebra.
On the other hand, Wikipedia gives a few possible definitions for the essential spectrum $\sigma_{ess}$ of an operator, some of which are mentioned to be invariant under compact perturbations.
Let $T \in \mathcal{L} (E)$. Is $\sigma (\pi(T))$ equal to $\sigma_{ess, k} (T)$ (see the Wikipedia page for the definition) for some $1 \leq k \leq 4$?
If so, which $k$?
Otherwise, do we still have $\rho_{ess} (T) = \rho(\pi(T))$, where $\rho$ (resp. $\rho_{ess}$) is the spctral radius (resp. the essential spectral radius)?
Motivation: I've been working with some families of quasi-compact operators for some time now. I've never seen the essential spectrum or essential spectral radius presented this way. If this works, it could make for a pretty nice introduction to the subject if I ever need to teach it. Oh, and unfortunately I don't have the references on the Wikipedia page at hand now, so I can't check them.
So, I couldn't find Spectral theory and differential operators by D.E. Edmunds and W.D. Evans, which mysteriously vanished from the library. However, a search on related questions on MathOverflow led me to Semi-Fredholm operators, perturbation theory and localized SVEP by P.Aliena.
The answer ot my question is on p. 131 (p.141 of the .pdf): the spectrum of $\pi (T) \in \mathcal{L} (E) / \mathcal{K} (E)$ is exactly the set of $\lambda \in \mathbb{C}$ such that $T-\lambda I$ is not Fredholm, that is, $\sigma_{ess,3} (T)$ using the notation of Wikipedia.