Let $L$ and $L_0$ be unbounded self-adjoint operators on $L^2(\mathbb{R})$ such as $L$ is a compact perturbation of $L_0$. I was wondering if it is possible to deduce an upper or lower bound on the dimension of the null space of $L$ from the dimension of the null space of $L_0$?
In fact, I am having trouble understanding the proof of lemma $4.3$, found in the article "Nonlinear stability of Mkdv breathers" by M. Alejo and C. Munoz.