Sharp estimates for singular values of Hankel operators

Abstract: We consider compact Hankel operators realized in $\ell^2(\mathbb Z_+)$ as infinite matrices $\Gamma$ with matrix elements $h(j+k)$. Roughly speaking, we show that, for all $\alpha>0$, the singular values $s_{n}$ of $\Gamma$ satisfy the bound $s_{n}= O(n^{-\alpha})$ as $n\to \infty$ provided $h(j)= O(j^{-1}(\log j)^{-\alpha})$ as $j\to \infty$. These estimates on $s_{n}$ are sharp in the power scale of $\alpha$. Similar results are obtained for Hankel operators $\mathbf\Gamma$ realized in $L^2(\mathbb R_+)$ as integral operators with kernels $\mathbf h(t+s)$. In this case the estimates of singular values of $\mathbf\Gamma$ are determined by the behavior of $\mathbf h(t)$ as $t\to 0$ and as $t\to\infty$.