Eigenvalue asymptotics for a class of multi-variable Hankel matrices

Abstract

A one-variable Hankel matrix Ha{H}_{a} is an infinite matrix Ha=[a(i+j)]i,j≥0{H}_{a}={\left[a\left(i+j)]}_{i,j\ge 0}. Similarly, for any d≥2d\ge 2, a dd-variable Hankel matrix is defined as Ha=[a(i+j)]{H}_{{\bf{a}}}=\left[{\bf{a}}\left({\bf{i}}+{\bf{j}})], where i=(i1,…,id){\bf{i}}=\left({i}_{1},\ldots ,{i}_{d}) and j=(j1,…,jd){\bf{j}}=\left({j}_{1},\ldots ,{j}_{d}), with i1,…,id,j1,…,jd≥0{i}_{1},\ldots ,{i}_{d},{j}_{1},\ldots ,{j}_{d}\ge 0. For γ>0\gamma \gt 0, Pushnitski and Yafaev proved that the eigenvalues of the compact one-variable Hankel matrices Ha{H}_{a} with a(j)=j−1(logj)−γa\left(j)={j}^{-1}{\left(\log j)}^{-\gamma }, for j≥2j\ge 2, obey the asymptotics λn(Ha)∼Cγn−γ{\lambda }_{n}\left({H}_{a})\hspace{0.33em} \sim \hspace{0.33em}{C}_{\gamma }{n}^{-\gamma }, as n→+∞n\to +\infty , where the constant Cγ{C}_{\gamma } is calculated explicitly. This article presents the following dd-variable analogue. Let γ>0\gamma \gt 0 and a(j)=j−d(logj)−γa\left(j)={j}^{-d}{\left(\log j)}^{-\gamma }, for j≥2j\ge 2. If a(j1,…,jd)=a(j1+⋯+jd){\bf{a}}\left({j}_{1},\ldots ,{j}_{d})=a\left({j}_{1}+\cdots +{j}_{d}), then Ha{H}_{{\bf{a}}} is compact and its eigenvalues follow the asymptotics λn(Ha)∼Cd,γn−γ{\lambda }_{n}\left({H}_{{\bf{a}}})\hspace{0.33em} \sim \hspace{0.33em}{C}_{d,\gamma }{n}^{-\gamma }, as n→+∞n\to +\infty , where the constant Cd,γ{C}_{d,\gamma } is calculated explicitly.

Keywords

• eigenvalues
• multi-variable hankel operators
• pseudo-differential operators
• schatten classes
• 47b35
• 47b06