Functional Analysis, infimum of $A+K$ where $K\in K(X)$

Question

This is a problem from Martin Schechter's Book (Principles of Functional Analysis)

If $X$ is a Banach Spaces and $A\in B(X)$, let $$|A|_K=\inf_{K\in K(X)}\|A+K\|.$$ Show that $|A|_K<1$ implies that $R(I-A)$ is closed in $X$ and $\dim N(I-A)=\dim N(I-A')<\infty$.

Answer 1

If $|A|_K < 1$, then $\exists K \in K(X)$ such that $\|A+K\| < 1$, which implies that $I - (A+K)$ is invertible in $B(X)$. In particular, $I-A$ is invertible in $B(X)/K(X)$. Atkinson's theorem tells you that $I-A$ is Fredholm - which is precisely what you want.