Let $A : X\to Y$ and $B : Y \to X$ be bounded operators between Banach spaces $X,Y$. Assume that $AB : Y\to Y$ is Fredholm. I would like to prove that $A(X)$ and $B(Y)$ are closed. Furthermore, possibly conclude that $A$ is also Fredholm.
I made some progress on it. For instant, since $\dim \ker AB = \dim B(Y)\cap \ker A$, then this intersection has finite dimension. It would be very nice to conclude that $\ker A$ is finite dimensional. However, I don't see how it is possible from here. But note that if $B$ is Fredholm, then necessarily $\dim B(Y) = \infty$, and hence, $\dim \ker A < \infty$.
On the other hand, if I show that $A(X)$ is closed, then since $\infty > \dim Y/AB(Y) \geq \dim Y/A(X)$ it would follow that $A$ is Fredholm.
However, I was not able to show that $A(X)$ and $B(Y)$ are closed, neither that is the case that $\dim B(Y) = \infty$ (in general).