Is essential spectrum not relevant to the topology on it Consider $F: \mathcal{D}(F)\subset X\rightarrow X$, we can define the essential spectrum as the set $\{\lambda\}$ s.t. the Fredholm index of $\lambda-F$ is not zero. Fredholm index can be written as
$ind\  \lambda-F= dim Ker(\lambda-F)-codim\  ran(\lambda-F)$ , where these two terms are both algebraic concept, does that mean essential spectrum is not related to the topology on the space?
 A: At least as a place-holder answer, in light of some comments: the usual definition/basic properties (whether something is part of the definition, or a basic property, depends on one's choice of logical order, and there is not a unique such...) of Fredholm operators on Banach spaces certainly does use the fact that the ambient space is a Banach space. This does imply that certain properties are equivalent to others, etc.
(Some properties of compact and/or Fredholm operators still make useful sense on "nuclear spaces", but things do start to unravel... I myself do know a little about this sort of extension, but mostly enough to know that spectral theory mostly doesn't work well... I remember Prof. Charles McCarthy, who got his PhD at Yale in the hey-day of operator theory there, once telling me that people spent a lot of time and effort trying to make spectral theory work more generally, but that it mostly just did not.)
It is certainly possible to choose some defining collection (not uniquely determined!) for Fredholm operators on Banach spaces, and use the same terminology in an arbitrary TVS or algebraic vector space.
Since the most-useful aspects of Fredholm and/or compact operators are (so far as I know) correct and easily provable on Banach spaces, and mostly fail otherwise, I myself am not aware of a useful definition beyond that case.
