Spectral inverse problems for compact Hankel operators

Abstract

Given two arbitrary sequences $(\lambda_j)_{j\ge 1}$ and $(\mu_j)_{j\ge 1}$ of real numbers satisfying $$|\lambda_1|>|\mu_1|>|\lambda_2|>|\mu_2|>...>| \lambda_j| >| \mu_j| \to 0\ ,$$ we prove that there exists a unique sequence $c=(c_n)_{n\in\Z_+}$, real valued, such that the Hankel operators $\Gamma_c$ and $\Gamma_{\tilde c}$ of symbols $c=(c_{n})_{n\ge 0}$ and $\tilde c=(c_{n+1})_{n\ge 0}$ respectively, are selfadjoint compact operators on $\ell^2(\Z_+)$ and have the sequences $(\lambda_j)_{j\ge 1}$ 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 $\Gamma_c$ and of $\Gamma_{\tilde c}$ 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 $$\rho_1>\sigma_1>\rho_2>\sigma_2>...> \rho_j> \sigma_j \to 0\ ,$$ we describe the set of sequences $c=(c_n)_{n\in\Z_+}$ of complex numbers such that the Hankel operators $\Gamma_c$ and $\Gamma_{\tilde c}$ are compact on $\ell ^2(\Z_+)$ and have sequences $(\rho_j)_{j\ge 1}$ and $(\sigma_j)_{j\ge 1}$ respectively as non zero singular values.