I already asked this on "Mathematics", but I guess (at least) Question 2 could be appropriate here as well.


Let's consider a selfadjoint compact operator $C\colon L^2[0,1] \rightarrow L^2[0,1],$ where $(L^2[0,1], ||\cdot||\_{L^2[0,1]})$ stands for the hilbert space of measurable function $f\colon [0,1] \rightarrow \mathbb{R}$ with $\int\_{[0,1]}f^2(x)\,\textrm{d}\lambda(x) < \infty$ with Lebesgue measure $\lambda.$ 


**Question 1:** What conditions $C$ has to fulfill such that the eigenvalues $(\lambda\_j)\_{j \in \mathbb{N}}$ of $C$ have multiplicity of one, meaning $\lambda\_i \neq \lambda\_j$ if $i \neq j$ with $i,j \in \mathbb{N}?$


**Question 2:** For compact operators $C$ we know that the sequence of eigenvalues $(\lambda\_j)\_{j \in \mathbb{N}}$ of $C$ belongs to $c\_0$ and we know $(\lambda\_j)\_{j \in \mathbb{N}} \in \ell^2$ if $C$ is Hilbert Schmidt, $(\lambda\_j)\_{j \in \mathbb{N}} \in \ell^1$ if $C$ is nuklear and $(\lambda\_j)\_{j \in \mathbb{N}} \in d$ if $C$ has finite image. But do I know something about the explicit decay rate of the sequence of eigenvalues of $C$ if $C$ fulfills certain properties? - meaning e.g. $\lambda\_j \sim j^{-3}$ for $j \rightarrow \infty$ or $\lambda\_j \sim r^j$ with $r \in (0,1)?$ 



Multiplicity of eigenvalues of a compact operator and explicit decay rate of the eigenvalues
I doubt that you'll find a very general condition that requires multiplicity $1$, other than those that are essentially restatements of that fact.
There will be conditions related to particular forms of operators, e.g. 
 Sturm-Liouville integral operators. 


Similarly, for explicit decay rates
you'll want something more explicit than one of these general classes.


