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 $t\to\infty$.
Categories: math.SP