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.
Best Answer
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.