Showing that the inverse of the perturbation of a compact operator by a bounded operator remains compact.

Question

The title says it all. If we have a Hilbert space $H$, then if $B\in \mathcal B(H)$, $L$ is a linear operator that is not necessarily bounded, $L^{-1}$ is compact, and $0\in \rho(L)\cap\rho(L+B)$, then show $(L+B)^{-1}$ is compact. Some ideas I had was to appeal to the fact that $K(H)$ (set of compact operators on $H$) is a two-sided ideal, but everything I tried only worked if $L$ was bounded (For example, I wanted to try the inverse bounded theorem). Any thoughts?

Answer 1

Hint: $(L+B)^{-1} = (I+L^{-1}B)^{-1} L^{-1}$.

EDIT: Further hint: the product of a bounded operator and a compact operator is compact.

FURTHER EDIT: What does the Fredholm alternative say about $(I + L^{-1} B)^{-1}$?