Behaviour of the spectrum of a compact operator w.r.t. perturbations. Suppose $A$ and $B$ are linear compact operators on a Hilbert space with $\sigma(A)$ and $\sigma(B)$ as their spectrum. 
Is it possible to obtain some continuity result of $\sigma(A+\epsilon B)$ as $\epsilon\downarrow 0$ towards $\sigma(A)$? Is the limiting behavior of the form "$\sigma(A)+\epsilon \sigma(B)$''?
Thanks in advance!

A partial answer. I think it is true if $\sigma(A)=\{0\}$. For $\lambda\neq 0$, $A-\lambda I$ is invertible. However, the set of isomorphisms is open in $\mathcal{B}(H)$, so there exist $\varepsilon>0$ such that $A-\lambda I+\varepsilon B$ is also invertible. This means that $\lambda\notin\sigma(A+\varepsilon B)$, for $\varepsilon$ small enough. This should imply that $\sigma(A+\varepsilon B)\to 0$.
