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Inverse spectral problems for compact Hankel operators

Part of: Infinite-dimensional Hamiltonian systems Special classes of linear operators

Published online by Cambridge University Press:  18 April 2013

Patrick Gérard  and
Sandrine Grellier
Patrick Gérard
Affiliation:
Université Paris-Sud XI, Laboratoire de Mathématiques d’Orsay, CNRS, UMR 8628, France Institut Universitaire de France, France ([email protected])
Sandrine Grellier
Affiliation:
Fédération Denis Poisson, MAPMO-UMR 6628, Département de Mathématiques, Université d’Orleans, 45067 Orléans Cedex 2, France ([email protected])
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Abstract

Given two arbitrary sequences $({\lambda }_{j} )_{j\geq 1} $ and $({\mu }_{j} )_{j\geq 1} $ of real numbers satisfying

$$\begin{eqnarray*}\displaystyle \vert {\lambda }_{1} \vert \gt \vert {\mu }_{1} \vert \gt \vert {\lambda }_{2} \vert \gt \vert {\mu }_{2} \vert \gt \cdots \gt \vert {\lambda }_{j} \vert \gt \vert {\mu }_{j} \vert \rightarrow 0, &&\displaystyle\end{eqnarray*}$$
we prove that there exists a unique sequence $c= ({c}_{n} )_{n\in { \mathbb{Z} }_{+ } } $, real valued, such that the Hankel operators ${\Gamma }_{c} $ and ${\Gamma }_{\tilde {c} } $ of symbols $c= ({c}_{n} )_{n\geq 0} $ and $\tilde {c} = ({c}_{n+ 1} )_{n\geq 0} $, respectively, are selfadjoint compact operators on ${\ell }^{2} ({ \mathbb{Z} }_{+ } )$ and have the sequences $({\lambda }_{j} )_{j\geq 1} $ and $({\mu }_{j} )_{j\geq 1} $, respectively, as non-zero eigenvalues. Moreover, we give an explicit formula for $c$ and we describe the kernel of ${\Gamma }_{c} $ and of ${\Gamma }_{\tilde {c} } $ in terms of the sequences $({\lambda }_{j} )_{j\geq 1} $ and $({\mu }_{j} )_{j\geq 1} $. More generally, given two arbitrary sequences $({\rho }_{j} )_{j\geq 1} $ and $({\sigma }_{j} )_{j\geq 1} $ of positive numbers satisfying
$$\begin{eqnarray*}\displaystyle {\rho }_{1} \gt {\sigma }_{1} \gt {\rho }_{2} \gt {\sigma }_{2} \gt \cdots \gt {\rho }_{j} \gt {\sigma }_{j} \rightarrow 0, &&\displaystyle\end{eqnarray*}$$
 we describe the set of sequences $c= ({c}_{n} )_{n\in { \mathbb{Z} }_{+ } } $ of complex numbers such that the Hankel operators ${\Gamma }_{c} $ and ${\Gamma }_{\tilde {c} } $ are compact on ${\ell }^{2} ({ \mathbb{Z} }_{+ } )$ and have sequences $({\rho }_{j} )_{j\geq 1} $ and $({\sigma }_{j} )_{j\geq 1} $, respectively, as non-zero singular values.

Keywords

Hankel operatorsinverse spectral problemscompressed shift operatoraction–angle variables

MSC classification

Secondary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators 37K15: Integration of completely integrable systems by inverse spectral and scattering methods
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 2 , April 2014 , pp. 273 - 301
Copyright
©Cambridge University Press 2013 

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