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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!

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I fixed some latex (makebox doesn't seem to work here). But you might clarify your question a little bit. I see two ways of interpreting it: 1) You are asking whether $ST$ is upper semi-Fredholm if $S$ and $T$ are, and whether the index of the product is the sum of the indices. OR 2) You are assuming that $S$, $T$, and $ST$ are upper semi-Fredholm, and you are only asking about the property of the index. (I think you are asking question 1, but it's not clear as written.) – MTS Nov 19 '12 at 5:20
Thanks, MTS. I means the first case. – Qingping Zeng Nov 29 '12 at 1:33

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.

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