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 $\sigma_e(T):=\sigma(\pi(T))$.
Let $\{e_n\}$ be the orthonormal basis for $\mathcal{H}$ and $\{\alpha_n\} \in \ell^{\infty}$.
Define $Te_n=\alpha_n e_n$.
Find the essential spectrum $\sigma_e(T)$ of the operator $T$.
What I've done:
I showed that $\sigma_e(T)\subseteq\sigma(T)$.
Also, I know that by Putnam-Schecheter, we have $\sigma(T)= \sigma_e \cup \Omega$, where $\Omega$ consists of some bounded components for the resolvent$\rho (\pi (T))$ and a sequence of isolated points in $\rho(\pi(T))$ converging to $\sigma_e(T)$.
Now I'm not sure how to continue. I think I need to use compactness at some point.
Any help is appreciated!
Thank you!
I suggest you try to compute the ordinary spectrum first. You can do that by calculating $(T-z)^{-1}$ explicitly. Then for $z$ in the spectrum try to construct an operator $S$ such that $S(T-z)-1$ and $(T-z)S-1$ are compact (you can think about finite rank first; it may be handy in the full solution that compact operator may be approximated by finite rank operators).
There exists also a more direct way which requires additional knowledge. The essential spectrum can be related to the notion of a Fredholm operator and its index. In your case it is fairly easy to figure out when $T-z$ is Fredholm, provided that you know required definitions.