Sharp Estimates for Singular Values of Hankel Operators

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• 作者：Alexander Pushnitski ; Dmitri Yafaev
• 关键词：Hankel operators ; discrete and continuous representations ; singular values ; Schatten classes
• 刊名：Integral Equations and Operator Theory
• 出版年：2015
• 出版时间：November 2015
• 年：2015
• 卷：83
• 期：3
• 页码：393-411
• 全文大小：626 KB
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• 作者单位：Alexander Pushnitski (1)
  Dmitri Yafaev (2)
  
  1. Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UK
  2. Department of Mathematics, University of Rennes-1, Campus Beaulieu, 35042, Rennes, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8989

文摘

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 \({t \to \infty}\) .