Let $\mathcal{H}$ be a complex, separable, infinite-dimensional Hilbert space. Given an operator $T\in\mathcal{B(H)}$ and $\pi:\mathcal{B(H)}\rightarrow\mathcal{B(H)}/\mathcal{K(H)}$ the canonical quotient map, we define the essential spectrum of $T$ by $\sigma_e(T):=\sigma(\pi(T))$. It is easy to see that $\sigma_e(T)\subseteq\sigma(T)$.
I know that if $T$ is a normal operator, then $\sigma_e(T)$ contains exactly the elements of $\sigma(T)$ that are not isolated eigenvalues of finite multiplicity. In contrast, a lot can change when $T$ is not normal. The forward unilateral shift $S:\ell^2(\mathbb{N})\rightarrow\ell^2(\mathbb{N})$ has spectrum $\sigma(S)=\overline{\mathbb{D}}$ (the closed unit disk) but essential spectrum $\sigma_e(S)=\mathbb{T}$ (the unit circle).
What I would like to know specifically is which elements in $\sigma(T)$ may disappear in $\sigma_e(T)$ when $T$ is a non-normal element in $\mathcal{B(H)}$. Does the essential spectrum always contain the boundary of the spectrum? What happens when $\sigma(T)$ has empty interior or is totally disconnected? If $\sigma(T)=\overline{\mathbb{D}}$ can multiple holes appear in $\sigma_e(T)$? What can be said about the spectral radius of $\pi(T)$ in terms of that of $T$?
A result due to Putnam and Schechter states the following: if $\mathcal{H}$ is a Hilbert space and $T\in\mathcal{B(H)}$, then for $\lambda\in\partial(\sigma(T))$ we have that $\lambda$ is isolated in $\sigma(T)$ or $\lambda\in\sigma_e(T)$. As a consequence we have that $\sigma(T)=\sigma_e(T)\cup\Omega$ where $\Omega$ is a (possibly empty) set consisting of some bounded components of $\sigma(T)$ and sequences of isolated points in $\sigma(T)$ converging to elements in $\sigma_e(T)$.
This gives us a better understanding of some of the situations above. In particular, if $\sigma(T)=\mathbb{T}$ (the unit circle) then no point of $\partial(\sigma(T))$ is isolated and hence $\sigma_e(T)=\mathbb{T}$ as well. As for an operator whose spectrum is $\overline{\mathbb{D}}$, it must be the case that the essential spectrum contains $\mathbb{T}$ but it could remove multiple holes in the disk or the whole interior.