Hostname: page-component-5cf477f64f-h6p2m Total loading time: 0 Render date: 2025-04-08T06:09:12.085Z Has data issue: false hasContentIssue false
  • English
  • Français

INVERSE SPECTRAL THEORY FOR A CLASS OF NON-COMPACT HANKEL OPERATORS

Part of: Special classes of linear operators Spaces and algebras of analytic functions

Published online by Cambridge University Press:  06 September 2018

Patrick Gérard  and
Alexander Pushnitski
Patrick Gérard
Affiliation:
Laboratoire de Mathématiques d’Orsay, Université Paris-Sud XI, CNRS, UMR 8628, 91405 Paris, France email [email protected]
Alexander Pushnitski
Affiliation:
Department of Mathematics, King’s College London, Strand, London WC2R 2LS, U.K. email [email protected]
Get access

Abstract

We characterize all bounded Hankel operators $\unicode[STIX]{x1D6E4}$ such that $\unicode[STIX]{x1D6E4}^{\ast }\unicode[STIX]{x1D6E4}$ has finite spectrum. We identify spectral data corresponding to such operators and construct inverse spectral theory including the characterization of these spectral data.

MSC classification

Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators 30H10: Hardy spaces
Type
Research Article
Information
Mathematika , Volume 65 , Issue 1 , 2019 , pp. 132 - 156
Copyright
Copyright © University College London 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Gérard, P., Wave turbulence and complete integrability, lectures at Fields Institute, summer school on non-linear dispersive PDEs and inverse scattering, Fields Institute, August 2017.Google Scholar
Gérard, P. and Grellier, S., The cubic Szegő equation. Ann. Sci. Éc. Norm. Supér. (4) 43 2010, 761–810.Google Scholar
Gérard, P. and Grellier, S., Invariant tori for the cubic Szegő equation. Invent. Math. 187(3) 2012, 707–754.Google Scholar
Gérard, P. and Grellier, S., The cubic Szegő equation and Hankel operators. Astérisque 389 2017.Google Scholar
Gérard, P. and Grellier, S., Inverse spectral problems for compact Hankel operators. J. Inst. Math. Jussieu 13 2014, 273–301.Google Scholar
Gérard, P. and Pushnitski, A., An inverse problem for self-adjoint positive Hankel operators. Int. Math. Res. Not. IMRN 13 2015, 4505–4535.Google Scholar
Gérard, P. and Pushnitski, A., Weighted model spaces and Schmidt subspaces of Hankel operators. Preprint, 2018, arXiv:1803.04295.Google Scholar
Nikolski, N. K., Operators, Functions, and Systems: An Easy Reading, Vol. 1: Hardy, Hankel, and Toeplitz (Mathematical Surveys and Monographs 92 ), American Mathematical Society (Providence, RI, 2002).Google Scholar
Peller, V. V., Hankel Operators and their Applications, Springer (2003).Google Scholar
Pocovnicu, O., Explicit formula for the solution of the Szegő equation on the real line and applications. Discrete Contin. Dyn. Syst. A 31 2011, 607–649.Google Scholar
Megretskiĭ, A. V., Peller, V. V. and Treil, S. R., The inverse spectral problem for self-adjoint Hankel operators. Acta Math. 174(2) 1995, 241–309.Google Scholar