Let $H$ be a hilbert space $ (\varphi_j)_{j \in \mathbb{N}} \subseteq H$ a Rieszbasis and by $\tilde{\varphi_j}$ we denote the biorthogonal sequence. Meaning
$ \langle \varphi_j , \tilde{\varphi_k} \rangle = \delta_{j,k} $
Let $\tau_j > 0 $ be a sequence in $\ell^1(\mathbb{N})$. Define
$D: H \to H$
$D: f \mapsto \sum_{j=1}^\infty \tau_j \langle f , \tilde{\varphi_j} \rangle \varphi_j $
The Eigenvalues being $\lambda_j (D) = \tau_j$.
The Question is what are the singular values of D?
$s_j (D) = \sqrt{\lambda_j (D^* D) } $
What I got so far there exists an other equivalent scalar product on $H$ such that $D$ is self adjoint. Therefor $s_{j,2} (D) = \lambda_j (D)$. But this only helps if the definition of singular values is independent of the concrete scalar product.
Now the other ansatz is going via Approximation Numbers, there is the question under wich conditions does
$a_j (D) = \lambda_j (D)$
while $a_j (D) := \min \{ \vert \vert D - T \vert \vert \,\ \vert \,\ rank (T) = j-1 \} $, hold?