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!