Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 7 / 3 / g

Use the stated Fredholm definition of the essential spectrum, which applies to general bounded operators:For each $z$, part (f) applied to $L-zI$ says that $L-zI$ is Fredholm exactly when $L+K-zI$ is Fredholm. Taking the complement of these equivalent Fredholm conditions givesThis is the general Weyl theorem for bounded compact perturbations. Individual isolated eigenvalues can change; the Fredholm obstruction, and hence this essential spectrum, cannot.