Inverse spectral problems for compact Hankel operators

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.