TITLE: Fredholm theory on Fr\'echet spaces
QUESTION [8 upvotes]: Dear everybody,
In my study of the classial Fredholm theory on Banach spaces, I am interested in the corresponding Fredholm theory on Fr\'echet spaces. But it seems to me that there is
little research results in this aspect. 
Let $X$, $Y$ and $Z$ be Fr\'echet spaces. An operator $T \in L(X,Y)$
is said to be upper (resp. lower) semi-Fredholm, if its kernel space
$\mathrm{ker}(T)$ is finite dimensional and its range space $R(T)$ is closed
in $Y$. In this case, the index of $T$ is defined as
$\mathrm{ind}(T)=\dim \mathrm{ker}(T) - \dim Y/R(T)$.
Now, one of my question is: 
Suppose that $T \in L(X,Y)$ and $S \in L(Y,Z)$ are upper
semi-Fredholm, $ST$ is upper semi-Fredholm and $\mathrm{ind}(ST)=
\mathrm{ind}(S)+\mathrm{ind}(T)$? (It holds in Banach spaces case) 
If you have any opinions to this question or this topic, please communicate with me. 
Thanks!

REPLY [2 votes]: This book: Spectral Theory of Linear Differential Operators and Comparison Algebras
 By Heinz Otto Cordes (mostly available on google books)
Discusses this, as does the earlier:
Einige Klassen singularen Gleichungen, by S. Prössdorf, Akademie-Verlag,
Berlin; Mathematische Reihe, Band 46, Birkhâuser, Verlag, Basel and
Stuttgart, 1974, 353 pp.