Let $H$ be a hilbert space $S,N: H \to H$, where $S$ is a self adjoint operator and $ A \leq ||S|| \leq B$ for some $A,B > 0$, and $N$ is a nuclear operator with eigenvalues $\lambda_k$. The eigenvalues are counted with their multiplicity and sorted such that they are monotone.
Now the question can the eigenvalues $\mu_k$ of $SN$, be estimated by:
$A \vert \lambda_k \vert \leq \vert \mu_k \vert \leq B \vert \lambda_k \vert $,
to be more precise since it is not clear how the $\mu_k$ are sorted. Does a sequence $n: \mathbb{N} \to \mathbb{N}$ wich is bijective exist such that:
$A \vert \lambda_k \vert \leq \vert \mu_{n_k} \vert \leq B \vert \lambda_k \vert $
holds. Any literature recommandation is appreciated, right now I am hoping to find something in "Introduction to the Theory of Lineal Nonselfadjoint Operators in Hilbert Space" by Gohberg and Krein.