$X$, $Y$ are Banach spaces. Let $S \in L(X, Y)$, $T \in L(Y, X)$, where $L(X, Y)$ denotes the Banach algebra of bounded linear operators from $X$ to $Y$. If we have that $Id_Y - ST \in \mathbb{K}(Y)$ and that $Id_X - TS \in \mathbb{K}(X)$. Is it then possible to define Index(T) to be $dim \ker(ST) - dim \ker(TS)$?

The intuition of this definition is that this is true for the case when $X=Y$ is some Hilbert space and $T=S^*$, the adjoint of $S$. I don't know how (if possible) to generalize this to the more general case.

If it is not true in the most general case, can you give a counterexample and a sufficient condition? Thank you!

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