Essential spectrum

Remarks

Since every invertible operator is Fredholm, $\sigma_{ess}(T)$ is a subset of the spectrum $\sigma(T)$; it is compact. It does not change when a compact operator is added to $T$. On a Hilbert space, it is the spectrum of the image of $T$ in the quotient algebra of bounded operators by compact operators, by Atkinson's theorem.

Example: if $X$ is infinite-dimensional, then $1 \in \sigma_{ess}(I)$, since $I - I = 0$ has an infinite-dimensional kernel.

Non-example: for $D(x_n) = (x_n / n)$ on $\ell^2$, $1 \in \sigma(D)$ but $1 \notin \sigma_{ess}(D)$. The kernel of $D - I$ is spanned by $e_1$, and its image is the closed subspace $\set{x : x_1 = 0}$ of codimension $1$, so $D - I$ is Fredholm. In fact $\sigma_{ess}(D) = \set{0}$.

In Banach spaces several non-equivalent variants of the essential spectrum are in use. For a self-adjoint operator on a Hilbert space all the usual variants agree. Then $\lambda \in \sigma_{ess}(T)$ if and only if $\lambda$ is in $\sigma(T)$ and is not an isolated eigenvalue of finite multiplicity.