Suppose $K \in \mathcal{B}(X)$ is a compact linear operator, where $X$ is a Banach space. Suppose $I-K$ is either left or right invertible. Show that $I-K$ is invertible and $I-(I-K)^{-1}$ is a compact linear operator.
I am not sure how to proceed with this problem. Intuitively it seems to me this holds because the compact operators form a two-sided ideal in the space of bounded linear operators. Can someone please help me.