Spectral inverse problems for compact Hankel operators

Given two arbitrary sequences $({\lambda }_j{)}_{j\ge 1}$ and $({\mu }_j{)}_{j\ge 1}$ of real numbers satisfying $$\mathrm{\mid }{\lambda }_1\mathrm{\mid }>\mathrm{\mid }{\mu }_1\mathrm{\mid }>\mathrm{\mid }{\lambda }_2\mathrm{\mid }>\mathrm{\mid }{\mu }_2\mathrm{\mid }>\mathrm{.}\mathrm{.}\mathrm{.}>\mathrm{\mid }{\lambda }_j\mathrm{\mid }>\mathrm{\mid }{\mu }_j\mathrm{\mid }\to 0,$$ we prove that there exists a unique sequence $c=(c_n{)}_{n\in {\mathbb{Z}}_{+}}$, real valued, such that the Hankel operators ${\mathrm{\Gamma }}_c$ and ${\mathrm{\Gamma }}_{\tilde{c}}$ of symbols $c=(c_n{)}_{n\ge 0}$ and $\tilde c=(c_{n+1}){n\ge 0}$respectively,areselfadjointcompactoperatorson$\ell^2(\Z+)$andhavethesequences$(\lambda_j){j\ge 1}$and$(\mu_j){j\ge 1}$ respectively as non zero eigenvalues. Moreover, we give an explicit formula for $c$ and we describe the kernel of ${\mathrm{\Gamma }}_c$ and of ${\mathrm{\Gamma }}_{\tilde{c}}$ in terms of the sequences $({\lambda }_j{)}_{j\ge 1}$ and $({\mu }_j{)}_{j\ge 1}$. More generally, given two arbitrary sequences $({\rho }_j{)}_{j\ge 1}$ and $({\sigma }_j{)}_{j\ge 1}$ of positive numbers satisfying $${\rho }_1>{\sigma }_1>{\rho }_2>{\sigma }_2>\mathrm{.}\mathrm{.}\mathrm{.}>{\rho }_j>{\sigma }_j\to 0,$$ we describe the set of sequences $c=(c_n{)}_{n\in {\mathbb{Z}}_{+}}$ of complex numbers such that the Hankel operators ${\mathrm{\Gamma }}_c$ and ${\mathrm{\Gamma }}_{\tilde{c}}$ are compact on $\ell ^2(\Z_+)$andhavesequences$(\rho_j){j\ge 1}$and$(\sigma_j){j\ge 1}$ respectively as non zero singular values.