Suppose $X$ be an infinite dimensional Banach space. How to prove that: $A$ and $B$ are two Fredholm operator on $X$, if $\mathrm{index}(A)=\mathrm{index}(B)$, then there exists an invertible operator $C$ such that $A-BC$ is compact???
I know that if $T$ is a Fredholm operator with $\mathrm{index}(T)=0$ iff there exist an invertible operator A and a compact operator $K$ such that $T=A+K$.
