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!

The first question has an answer and is "yes", see Dunford-Schwartz, Linear operators, part II, XI-9, Lemma 5; it reads: 
"Let $A_n, A$ be compact operators and $A_n\rightarrow A$ in the uniform operator topology. Let $\lambda_m(A)$ be an enumeration of the non-zero eigenvalues of $A$, each repeated according to its multiplicity. Then, there exist enumerations $\lambda_m(A_n)$ of the non-zero eigenvalues of $A_n$, with repetitions according to multiplicity, such that $$\lim_{n\rightarrow\infty} \lambda_m(A_n)=\lambda_m(A), \quad m\geq 1,$$ the limit being uniform in $m$."
So, under compact perturbations $\epsilon B$ of a compact operator $A$, the perturbed spectrum moves continuously. I don't know about your second question, the order of convergence might be a little bit tricky to obtain, some Rellich result might be useful. Consider checking Kato's book "Perturbation Theory for Linear Operators". 
