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 \({{{\bf \Gamma}}}\) 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 \({t \to \infty}\) .
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References
Bergh J., Löfström J.: Interpolation Spaces. Springer, New York (1976)
Birman M.S., Solomyak M.Z.: Spectral Theory of Selfadjoint Operators in Hilbert Space. D. Reidel, Dordrecht (1987)
Bonsall, F.F.: Some nuclear Hankel operators. In: Aspects of Mathematics and its Applications, pp. 227–238, North-Holland Math. Library 34. North-Holland, Amsterdam (1986)
Glover K., Lam J., Partington J.R.: Rational approximation of a class of infinite-dimensional systems I: singular values of Hankel operators. Math. Control Signals Syst. 3, 325–344 (1990)
Gohberg, I.C., Kreĭn, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space. Am. Math. Soc., Providence, Rhode Island (1970)
Parfenov O.G.: Estimates for singular numbers of Hankel operators. Math. Notes. 49, 610–613 (1994)
Peller V.: Hankel Operators and Their Applications. Springer, New York (2003)
Pushnitski, A., Yafaev, D.: Asymptotic behaviour of eigenvalues of Hankel operators. Int. Math. Res. Notices. doi: 10.1093/imrn/rnv048; arXiv:1412.2633
Widom H.: Hankel matrices. Trans. Am. Math. Soc. 121(1), 1–35 (1966)
Yafaev D.R.: Criteria for Hankel operators to be sign-definite. Anal. PDE 8(1), 183–221 (2015)
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Pushnitski, A., Yafaev, D. Sharp Estimates for Singular Values of Hankel Operators. Integr. Equ. Oper. Theory 83, 393–411 (2015). https://doi.org/10.1007/s00020-015-2239-0
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DOI: https://doi.org/10.1007/s00020-015-2239-0