Asymptotic behaviour of eigenvalues 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 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 $t\to \infty$.

References in corpus (2)

• Quasi-diagonalization of Hankel operators
• Sharp estimates for singular values of Hankel operators