Given a Hilbert space $\mathcal{H}$, what are the eigenvalues of $(T^*T)^{1/2}$ in terms of the eigenvalues of a compact operator $T: \mathcal{H} \rightarrow \mathcal{H},$ where $T^*$ denotes the adjoint of $T$?
I want to contrast a result stated in terms of the eigenvalues of $(T^*T)^{1/2}$ to another result stated in terms of the eigenvalues of $T$. I am mainly interested in expressing the convergence of $\sum_k (\frac{1}{\lambda_k})^{-\nu}$ (where $\lambda_k$ is the $k-$th eigenvalue of $(T^*T)^{1/2}$ in descending order and $\nu$ is a real number) in terms of the eigenvalues of $T$.