Reference for invariance of essential spectrum under relatively compact perturbations

Question

I'm looking for a proof of the following statement:

Let $X$ be a Banach space and $T$ be a closed map on $X$. For any relatively compact map $A$ the essential spectrum of $T$ and $T+A$ are the same.

This is proven in Kato's Perturbation Theory in IV - Stability Theorems- Section 5. The proof goes via the theory of semi-Fredholm operators, however I would like to know of any other references/proof techniques/books I can look at to see this proof. (I don't like Kato.)

The corresponding result in Hilbert spaces (Weyls theorem) I've seen several proofs for, so I am looking for the result for Banach spaces.

Answer 1

As an answer to this question (which I spent quite a while longer looking for) I found the following reference: http://evm.ivic.gob.ve/libropaiena.pdf, a book written by Pietro Aiena.

This book is wonderful: and if anyone in the future finds this question I highly recommend it. It contains several possible generalizations of Weyl's theorem to the Banach case, eg, through Fredholm indices, and he contains a lengthy discussion on the relation between Weyl's theorem and Brouwders theorem. It also has lengthy references which provide an excellent starting point for any further questions on the topic/if the original papers are needed.

Answer 2

Claim: If $T$ is a closed Fredholm operator acting on a complex Banach space $X$ and $A$ is a relatively $T$-compact operator, then the fact that $T+A$ is Fredholm can be derived from the stability under compact perturbations of the bounded Fredholm operators:

Indeed, denoting $X_T$ the domain $D(T)$ with the graph norm $\|x\|_T :=\|x\| +\|Tx\|$, and $j_T$ the inclusion on $X_T$ into $X$, then $Tj_T$ is a bounded Fredholm operator and $Aj_T$ is a compact operator (note that $D(T)\subset D(A)$). Thus $(T+A)j_T$ is bounded Fredholm, hence $T+A$ Fredholm.

The essential spectrum of $T$ is $\sigma_e(T):=\{\lambda : T-\lambda I \textrm{ Fredhom }\}$. The fact that $\sigma_e(T+A)=\sigma_e(T)$ is obtained by applying the Claim to $T-\lambda I$.

In S. Goldberg, Unbounded linear operators, McGraw-Hill 1966, you can find additional details.