Assume $X$ is a Banach space and that $T$ is a bounded linear map and $K$ is a compact linear map from $X$ to itself.
I need to prove that that if $\lambda$ is in the spectrum of $T$ but is not an eigenvalue of finite multiplicity then $\lambda$ is in the spectrum of $T+K$. I know some properties of the spectrum of a compact operator, but I am not sure what to say about the sum of a bounded and a compact one.
For instance, for a compact operator we know that every $\lambda \neq 0$ in the spectrum has finite multiplicity and is an eigenvalue. But since the sum $T+K$ is not compact in general we can not say much about them at first sight. Also, I know that $T+K$ is Fredholm, but how could this be useful?
I would appreciate any hints on the problem. Thanks!
Best Answer
The Fredholm aspect is crucial here.
The condition of not being an eigenvalue of finite multiplicity guarantees that $\lambda$ is in the essential spectrum of $T$.
The essential spectrum does not change under compact perturbations.