Asymptotic behaviour of eigenvalues of Hankel operators

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues of Hankel operators, showing how their decay rates relate to the asymptotic form of the matrix elements or kernels, with explicit formulas for the coefficients.

Contribution

It provides explicit asymptotic formulas for eigenvalues of Hankel operators in both discrete and continuous settings based on the asymptotic behavior of their defining functions.

Findings

Eigenvalues decay as n^{-α} for certain asymptotic forms of matrix elements.

Explicit expressions for asymptotic coefficients in terms of the functions' parameters.

Results apply to both discrete and integral 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 if $h(j)\sim (b_{1}+ (-1)^j b_{-1}) j^{-1}(\log j)^{-\alpha}$ as $j\to \infty$ for some $\alpha>0$, then the eigenvalues of $\Gamma$ satisfy $\lambda_{n}^{\pm} (\Gamma)\sim c^{\pm} n^{-\alpha}$ as $n\to \infty$. The asymptotic coefficients $c^{\pm}$ are explicitly expressed in terms of the asymptotic coefficients $b_{1} $ and $b_{-1}$. 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 asymptotics of eigenvalues $\lambda_{n}^{\pm} (\mathbf \Gamma)$ are determined by the behaviour of $\mathbf h(t)$ as $t\to 0$ and as $t\to \infty$.