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 ${{{\bf \Gamma}}}$ realized in ${L^2({\mathbb{R}}_+)}$ as integral operators with kernels ${\mathbf{h}(t+s)}$ . In this case the estimates of singular values of ${{{\bf \Gamma}}}$ are determined by the behavior of ${\mathbf{h}(t)}$ as ${t \to 0}$ and as ${t \to \infty}$ .