Sharp estimates for singular values of Hankel operators

TL;DR

This paper establishes sharp decay rates for the singular values of compact Hankel operators, linking their asymptotic behavior to the decay of their matrix elements or kernels, with results applicable in both discrete and continuous settings.

Contribution

It provides precise decay estimates for singular values of Hankel operators based on the decay of their defining functions, extending known results with sharp bounds.

Findings

Singular values decay as O(n^{-eta}) for specific decay rates of matrix elements.

Results are sharp in the power scale of decay parameter lpha.

Analogous estimates are obtained for integral Hankel operators in L^2(7).

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 $t\to\infty$.