I've asked this problem in MSE several days ago, see here. But there is no reply up until now. Maybe I wrote things too complicated there and so I'll write a very clean problem here. For background intuitions and the long story behind the problem, interested people can read the original MSE problem.
Problem. Let $X$ be a complex Banach space and $A$ be a densely defined closed operator from $D(A)\subset X$ to $X$. Define the discrete spectrum $\sigma_{\text{d}}(A)$ of $A$ by $$ \sigma_{\text{d}}(A):=\{\lambda\in\mathbb C: \lambda \text{ is an isolated eigenvalue of }\ A \text{ and has finite algebraic multiplicity}\}.$$ Here the algebraic multiplicity of an isolated eigenvalue $\lambda$ is defined as the dimension of the image of the Riesz projection $P_\lambda$ defined by $$P_\lambda=\frac{1}{2\pi i}\oint_\gamma (z-A)^{-1}\,dz,$$ where $\gamma$ is any simple closed curve around $\lambda$ which only contains one element of $\sigma(A)$ inside the curve. Now define $$\sigma_{\text{ess}}(A):=\sigma(A)\setminus\sigma_{\text{d}}(A).$$ Show that $\sigma_{\text{ess}}(A+C)=\sigma_{\text{ess}}(A)$ for any bounded compact operator $C$ on $X$.
We usually define the essential spectrum in the above way for self-adjoint operators on Hilbert spaces, and then we can use Weyl's criterion to show the stability of the essential spectrum under (relatively) compact perturbations. However, for general closed operators, people usually define the essential spectrum through Fredholm or semi-Fredholm operators. I just wonder whether the essential spectrum defined analogous to self-adjoint case is stable under compact perturbations.
Any hints or useful references are welcome!
